Theorem infxpidmlem3 4935

Description: Lemma for infxpidm 4945. A sufficient condition for a set B to belong to H.

Hypotheses

Ref	Expression
infxpidmlem.1	⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}
infxpidmlem2.2	⊢ B ∈ V
infxpidmlem3.3	⊢ D ∈ V

Assertion

Ref	Expression
infxpidmlem3	⊢ (((ω ≼ D ∧ D ⊆ A) ∧ B:(D × D)–1-1-onto→D) → B ∈ H)

Distinct variable group(s):   t,f,A   B,f,t

Proof of Theorem infxpidmlem3

Step	Hyp	Ref	Expression
1		infxpidmlem3.3	. . 3 ⊢ D ∈ V
2		breq2 2066	. . . . 5 ⊢ (x = D → (ω ≼ x ↔ ω ≼ D))
3		sseq1 1521	. . . . 5 ⊢ (x = D → (x ⊆ A ↔ D ⊆ A))
4	2, 3	anbi12d 476	. . . 4 ⊢ (x = D → ((ω ≼ x ∧ x ⊆ A) ↔ (ω ≼ D ∧ D ⊆ A)))
5		xpeq1 2440	. . . . . . 7 ⊢ (x = D → (x × x) = (D × x))
6		xpeq2 2441	. . . . . . 7 ⊢ (x = D → (D × x) = (D × D))
7	5, 6	eqtrd 1128	. . . . . 6 ⊢ (x = D → (x × x) = (D × D))
8		f1oeq2 2796	. . . . . 6 ⊢ ((x × x) = (D × D) → (B:(x × x)–1-1-onto→x ↔ B:(D × D)–1-1-onto→x))
9	7, 8	syl 12	. . . . 5 ⊢ (x = D → (B:(x × x)–1-1-onto→x ↔ B:(D × D)–1-1-onto→x))
10		f1oeq3 2797	. . . . 5 ⊢ (x = D → (B:(D × D)–1-1-onto→x ↔ B:(D × D)–1-1-onto→D))
11	9, 10	bitrd 406	. . . 4 ⊢ (x = D → (B:(x × x)–1-1-onto→x ↔ B:(D × D)–1-1-onto→D))
12	4, 11	anbi12d 476	. . 3 ⊢ (x = D → (((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x) ↔ ((ω ≼ D ∧ D ⊆ A) ∧ B:(D × D)–1-1-onto→D)))
13	1, 12	cla4ev 1401	. 2 ⊢ (((ω ≼ D ∧ D ⊆ A) ∧ B:(D × D)–1-1-onto→D) → ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x))
14		olc 224	. . 3 ⊢ (∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x) → (B = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x)))
15		infxpidmlem.1	. . . 4 ⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}
16		infxpidmlem2.2	. . . 4 ⊢ B ∈ V
17	15, 16	infxpidmlem2 4934	. . 3 ⊢ (B ∈ H ↔ (B = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x)))
18	14, 17	sylibr 175	. 2 ⊢ (∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x) → B ∈ H)
19	13, 18	syl 12	1 ⊢ (((ω ≼ D ∧ D ⊆ A) ∧ B:(D × D)–1-1-onto→D) → B ∈ H)

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  ωcom 2372   × cxp 2408  –1-1-onto→wf1o 2421   ≼ cdom 3272

This theorem is referenced by:  infxpidmlem8 4940  infxpidmlem10 4942  infxpidmlem11 4943

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

metamath.org