Theorem unfilem3 3440

Description: Lemma for proving that the union of two finite sets is finite.

Assertion

Ref	Expression
unfilem3	⊢ ((A ∈ ω ∧ B ∈ ω) → B ≈ ((A +o B) ∖ A))

Proof of Theorem unfilem3

Step	Hyp	Ref	Expression
1		f1oeq1 2795	. . . 4 ⊢ (f = {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} → (f:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A)))
2	1	cla4egv 1397	. . 3 ⊢ ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} ∈ V → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A) → ∃f f:B–1-1-onto→((A +o B) ∖ A)))
3		opreq1 3006	. . . . . . . . . . 11 ⊢ (A = if(A ∈ ω, A, ∅) → (A +o x) = (if(A ∈ ω, A, ∅) +o x))
4	3	cleq2d 1112	. . . . . . . . . 10 ⊢ (A = if(A ∈ ω, A, ∅) → (y = (A +o x) ↔ y = (if(A ∈ ω, A, ∅) +o x)))
5	4	anbi2d 468	. . . . . . . . 9 ⊢ (A = if(A ∈ ω, A, ∅) → ((x ∈ B ∧ y = (A +o x)) ↔ (x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))))
6	5	biopabdv 2102	. . . . . . . 8 ⊢ (A = if(A ∈ ω, A, ∅) → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} = {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))})
7		f1oeq1 2795	. . . . . . . 8 ⊢ ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} = {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))} → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((A +o B) ∖ A)))
8	6, 7	syl 12	. . . . . . 7 ⊢ (A = if(A ∈ ω, A, ∅) → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((A +o B) ∖ A)))
9		opreq1 3006	. . . . . . . . . 10 ⊢ (A = if(A ∈ ω, A, ∅) → (A +o B) = (if(A ∈ ω, A, ∅) +o B))
10	9	difeq1d 1587	. . . . . . . . 9 ⊢ (A = if(A ∈ ω, A, ∅) → ((A +o B) ∖ A) = ((if(A ∈ ω, A, ∅) +o B) ∖ A))
11		difeq2 1583	. . . . . . . . 9 ⊢ (A = if(A ∈ ω, A, ∅) → ((if(A ∈ ω, A, ∅) +o B) ∖ A) = ((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)))
12	10, 11	eqtrd 1128	. . . . . . . 8 ⊢ (A = if(A ∈ ω, A, ∅) → ((A +o B) ∖ A) = ((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)))
13		f1oeq3 2797	. . . . . . . 8 ⊢ (((A +o B) ∖ A) = ((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) → ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
14	12, 13	syl 12	. . . . . . 7 ⊢ (A = if(A ∈ ω, A, ∅) → ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
15	8, 14	bitrd 406	. . . . . 6 ⊢ (A = if(A ∈ ω, A, ∅) → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A) ↔ {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
16		eleq2 1150	. . . . . . . . . 10 ⊢ (B = if(B ∈ ω, B, ∅) → (x ∈ B ↔ x ∈ if(B ∈ ω, B, ∅)))
17	16	anbi1d 469	. . . . . . . . 9 ⊢ (B = if(B ∈ ω, B, ∅) → ((x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x)) ↔ (x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))))
18	17	biopabdv 2102	. . . . . . . 8 ⊢ (B = if(B ∈ ω, B, ∅) → {⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))} = {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))})
19		f1oeq1 2795	. . . . . . . 8 ⊢ ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))} = {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))} → ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
20	18, 19	syl 12	. . . . . . 7 ⊢ (B = if(B ∈ ω, B, ∅) → ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
21		f1oeq2 2796	. . . . . . 7 ⊢ (B = if(B ∈ ω, B, ∅) → ({⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅))))
22		opreq2 3007	. . . . . . . . 9 ⊢ (B = if(B ∈ ω, B, ∅) → (if(A ∈ ω, A, ∅) +o B) = (if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)))
23	22	difeq1d 1587	. . . . . . . 8 ⊢ (B = if(B ∈ ω, B, ∅) → ((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) = ((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅)))
24		f1oeq3 2797	. . . . . . . 8 ⊢ (((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) = ((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅)) → ({⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅))))
25	23, 24	syl 12	. . . . . . 7 ⊢ (B = if(B ∈ ω, B, ∅) → ({⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅))))
26	20, 21, 25	3bitrd 422	. . . . . 6 ⊢ (B = if(B ∈ ω, B, ∅) → ({⟨x, y⟩∣(x ∈ B ∧ y = (if(A ∈ ω, A, ∅) +o x))}:B–1-1-onto→((if(A ∈ ω, A, ∅) +o B) ∖ if(A ∈ ω, A, ∅)) ↔ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅))))
27		peano1 2390	. . . . . . . 8 ⊢ ∅ ∈ ω
28	27	elimel 1793	. . . . . . 7 ⊢ if(A ∈ ω, A, ∅) ∈ ω
29	27	elimel 1793	. . . . . . 7 ⊢ if(B ∈ ω, B, ∅) ∈ ω
30		cleqid 1102	. . . . . . 7 ⊢ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))} = {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}
31	28, 29, 30	unfilem2 3439	. . . . . 6 ⊢ {⟨x, y⟩∣(x ∈ if(B ∈ ω, B, ∅) ∧ y = (if(A ∈ ω, A, ∅) +o x))}:if(B ∈ ω, B, ∅)–1-1-onto→((if(A ∈ ω, A, ∅) +o if(B ∈ ω, B, ∅)) ∖ if(A ∈ ω, A, ∅))
32	15, 26, 31	dedth2h 1787	. . . . 5 ⊢ ((A ∈ ω ∧ B ∈ ω) → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A))
33		f1ofn 2801	. . . . 5 ⊢ ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}:B–1-1-onto→((A +o B) ∖ A) → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} Fn B)
34	32, 33	syl 12	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω) → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} Fn B)
35		fnex 2740	. . . . 5 ⊢ (B ∈ ω → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} Fn B → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} ∈ V))
36	35	adantl 305	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω) → ({⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} Fn B → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} ∈ V))
37	34, 36	mpd 46	. . 3 ⊢ ((A ∈ ω ∧ B ∈ ω) → {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} ∈ V)
38	2, 37, 32	sylc 62	. 2 ⊢ ((A ∈ ω ∧ B ∈ ω) → ∃f f:B–1-1-onto→((A +o B) ∖ A))
39		oprex 3018	. . . 4 ⊢ (A +o B) ∈ V
40		difexg 1703	. . . 4 ⊢ ((A +o B) ∈ V → ((A +o B) ∖ A) ∈ V)
41	39, 40	ax-mp 6	. . 3 ⊢ ((A +o B) ∖ A) ∈ V
42	41	bren 3282	. 2 ⊢ (B ≈ ((A +o B) ∖ A) ↔ ∃f f:B–1-1-onto→((A +o B) ∖ A))
43	38, 42	sylibr 175	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → B ≈ ((A +o B) ∖ A))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484  ∅c0 1707  ifcif 1776   class class class wbr 2054  {copab 2055  ωcom 2372   Fn wfn 2417  –1-1-onto→wf1o 2421  (class class class)co 3001   +_o coa 3101   ≈ cen 3271

This theorem is referenced by:  unfi 3441

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-en 3274

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