Theorem infxpidmlem2 4934

Description: Lemma for infxpidm 4945. A necessary and 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

Assertion

Ref	Expression
infxpidmlem2	⊢ (B ∈ H ↔ (B = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x)))

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

Proof of Theorem infxpidmlem2

Step	Hyp	Ref	Expression
1		infxpidmlem2.2	. . 3 ⊢ B ∈ V
2		cleq1 1107	. . . 4 ⊢ (f = B → (f = ∅ ↔ B = ∅))
3		f1oeq1 2795	. . . . . 6 ⊢ (f = B → (f:(t × t)–1-1-onto→t ↔ B:(t × t)–1-1-onto→t))
4	3	anbi2d 468	. . . . 5 ⊢ (f = B → (((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t) ↔ ((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t)))
5	4	biexdv 936	. . . 4 ⊢ (f = B → (∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t) ↔ ∃t((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t)))
6	2, 5	orbi12d 475	. . 3 ⊢ (f = B → ((f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t)) ↔ (B = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t))))
7		infxpidmlem.1	. . 3 ⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}
8	1, 6, 7	elab2 1419	. 2 ⊢ (B ∈ H ↔ (B = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t)))
9		breq2 2066	. . . . . 6 ⊢ (x = t → (ω ≼ x ↔ ω ≼ t))
10		sseq1 1521	. . . . . 6 ⊢ (x = t → (x ⊆ A ↔ t ⊆ A))
11	9, 10	anbi12d 476	. . . . 5 ⊢ (x = t → ((ω ≼ x ∧ x ⊆ A) ↔ (ω ≼ t ∧ t ⊆ A)))
12		xpeq1 2440	. . . . . . . 8 ⊢ (x = t → (x × x) = (t × x))
13		xpeq2 2441	. . . . . . . 8 ⊢ (x = t → (t × x) = (t × t))
14	12, 13	eqtrd 1128	. . . . . . 7 ⊢ (x = t → (x × x) = (t × t))
15		f1oeq2 2796	. . . . . . 7 ⊢ ((x × x) = (t × t) → (B:(x × x)–1-1-onto→x ↔ B:(t × t)–1-1-onto→x))
16	14, 15	syl 12	. . . . . 6 ⊢ (x = t → (B:(x × x<.I>)–1-1-onto→x ↔ B:(t × t)–1-1-onto→x))
17		f1oeq3 2797	. . . . . 6 ⊢ (x = t → (B:(t × t)–1-1-onto→x ↔ B:(t × t)–1-1-onto→t))
18	16, 17	bitrd 406	. . . . 5 ⊢ (x = t → (B:(x × x)–1-1-onto→x ↔ B:(t × t)–1-1-onto→t))
19	11, 18	anbi12d 476	. . . 4 ⊢ (x = t → (((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x) ↔ ((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t)))
20	19	cbvexv 973	. . 3 ⊢ (∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x) ↔ ∃t((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t))
21	20	orbi2i 214	. 2 ⊢ ((B = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x)) ↔ (B = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ B:(t × t)–1-1-onto→t)))
22	8, 21	bitr4 154	1 ⊢ (B ∈ H ↔ (B = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ B:(x × x)–1-1-onto→x)))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  ωcom 2372   × cxp   –1-1-onto→wf1o 2421   ≼ cdom 3272

This theorem is referenced by:  infxpidmlem3 4935  infxpidmlem4 4936  infxpidmlem7 4939  infxpidmlem8 4940  infxpidmlem12 4944

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

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