Theorem unfilem2 3439

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

Hypotheses

Ref	Expression
unfilem1.1	⊢ A ∈ ω
unfilem1.2	⊢ B ∈ ω
unfilem1.3	⊢ F = {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}

Assertion

Ref	Expression
unfilem2	⊢ F:B–1-1-onto→((A +o B) ∖ A)

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem unfilem2

Step	Hyp	Ref	Expression
1		oprex 3018	. . . . . . . . 9 ⊢ (A +o x) ∈ V
2		unfilem1.3	. . . . . . . . 9 ⊢ F = {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}
3	1, 2	fnopab2 2747	. . . . . . . 8 ⊢ F Fn B
4		unfilem1.1	. . . . . . . . 9 ⊢ A ∈ ω
5		unfilem1.2	. . . . . . . . 9 ⊢ B ∈ ω
6	4, 5, 2	unfilem1 3438	. . . . . . . 8 ⊢ ran F = ((A +o B) ∖ A)
7	3, 6	pm3.2i 234	. . . . . . 7 ⊢ (F Fn B ∧ ran F = ((A +o B) ∖ A))
8		df-fo 2436	. . . . . . 7 ⊢ (F:B–onto→((A +o B) ∖ A) ↔ (F Fn B ∧ ran F = ((A +o B) ∖ A)))
9	7, 8	mpbir 165	. . . . . 6 ⊢ F:B–onto→((A +o B) ∖ A)
10		fof 2788	. . . . . 6 ⊢ (F:B–onto→((A +o B) ∖ A) → F:B–→((A +o B) ∖ A))
11	9, 10	ax-mp 6	. . . . 5 ⊢ F:B–→((A +o B) ∖ A)
12		opreq2 3007	. . . . . . . . . 10 ⊢ (x = z → (A +o x) = (A +o z))
13		oprex 3018	. . . . . . . . . 10 ⊢ (A +o z) ∈ V
14	12, 2, 13	fvopab4 2871	. . . . . . . . 9 ⊢ (z ∈ B → (F ‘z) = (A +o z))
15		opreq2 3007	. . . . . . . . . 10 ⊢ (x = w → (A +o x) = (A +o w))
16		oprex 3018	. . . . . . . . . 10 ⊢ (A +o w) ∈ V
17	15, 2, 16	fvopab4 2871	. . . . . . . . 9 ⊢ (w ∈ B → (F ‘w) = (A +o w))
18	14, 17	cleqan12d 1116	. . . . . . . 8 ⊢ ((z ∈ B ∧ w ∈ B) → ((F ‘z) = (F ‘w) ↔ (A +o z) = (A +o w)))
19		nnacan 3184	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ z ∈ ω ∧ w ∈ ω) → ((A +o z) = (A +o w) ↔ z = w))
20	4, 19	mp3an1 639	. . . . . . . . 9 ⊢ ((z ∈ ω ∧ w ∈ ω) → ((A +o z) = (A +o w) ↔ z = w))
21		elnn 2383	. . . . . . . . . 10 ⊢ ((z ∈ B ∧ B ∈ ω) → z ∈ ω)
22	5, 21	mpan2 519	. . . . . . . . 9 ⊢ (z ∈ B → z ∈ ω)
23		elnn 2383	. . . . . . . . . 10 ⊢ ((w ∈ B ∧ B ∈ ω) → w ∈ ω)
24	5, 23	mpan2 519	. . . . . . . . 9 ⊢ (w ∈ B → w ∈ ω)
25	20, 22, 24	syl2an 349	. . . . . . . 8 ⊢ ((z ∈ B ∧ w ∈ B) → ((A +o z) = (A +o w) ↔ z = w))
26	18, 25	bitrd 406	. . . . . . 7 ⊢ ((z ∈ B ∧ w ∈ B) → ((F ‘z) = (F ‘w) ↔ z = w))
27	26	biimpd 135	. . . . . 6 ⊢ ((z ∈ B ∧ w ∈ B) → ((F ‘z) = (F ‘w) → z = w))
28	27	rgen2 1248	. . . . 5 ⊢ ∀z ∈ B ∀w ∈ B ((F ‘z) = (F ‘w) → z = w)
29	11, 28	pm3.2i 234	. . . 4 ⊢ (F:B–→((A +o B) ∖ A) ∧ ∀z ∈ B ∀w ∈ B ((F ‘z) = (F ‘w) → z = w))
30		f1fv 2916	. . . 4 ⊢ (F:B–1-1→((A +o B) ∖ A) ↔ (F:B–→((A +o B) ∖ A) ∧ ∀z ∈ B ∀w ∈ B ((F ‘z) = (F ‘w) → z = w)))
31	29, 30	mpbir 165	. . 3 ⊢ F:B–1-1→((A +o B) ∖ A)
32	31, 9	pm3.2i 234	. 2 ⊢ (F:B–1-1→((A +o B) ∖ A) ∧ F:B–onto→((A +o B) ∖ A))
33		df-f1o 2437	. 2 ⊢ (F:B–1-1-onto→((A +o B) ∖ A) ↔ (F:B–1-1→((A +o B) ∖ A) ∧ F:B–onto→((A +o B) ∖ A)))
34	32, 33	mpbir 165	1 ⊢ F:B–1-1-onto→((A +o B) ∖ A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ∖ cdif 1484  {copab 2055  ωcom 2372  ran crn 2411   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421   ‘cfv 2422  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  unfilem3 3440

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

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