Theorem infxpidmlem4 4936

Description: Lemma for infxpidm 4945. The domain of a member of H is the cross product of its range.

Hypothesis

Ref	Expression
infxpidmlem.1	⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}

Assertion

Ref	Expression
infxpidmlem4	⊢ (g ∈ H → dom g = (ran g × ran g))

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

Proof of Theorem infxpidmlem4

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . 3 ⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}
2		visset 1350	. . 3 ⊢ g ∈ V
3	1, 2	infxpidmlem2 4934	. 2 ⊢ (g ∈ H ↔ (g = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ g:(x × x)–1-1-onto→x)))
4		dmeq 2531	. . . . 5 ⊢ (g = ∅ → dom g = dom ∅)
5		dm0 2542	. . . . 5 ⊢ dom ∅ = ∅
6	4, 5	syl6eq 1140	. . . 4 ⊢ (g = ∅ → dom g = ∅)
7		rneq 2555	. . . . . . 7 ⊢ (g = ∅ → ran g = ran ∅)
8		rn0 2567	. . . . . . 7 ⊢ ran ∅ = ∅
9	7, 8	syl6eq 1140	. . . . . 6 ⊢ (g = ∅ → ran g = ∅)
10		xpeq2 2441	. . . . . 6 ⊢ (ran g = ∅ → (ran g × ran g) = (ran g × ∅))
11	9, 10	syl 12	. . . . 5 ⊢ (g = ∅ → (ran g × ran g) = (ran g × ∅))
12		xp0 2652	. . . . 5 ⊢ (ran g × ∅) = ∅
13	11, 12	syl6eq 1140	. . . 4 ⊢ (g = ∅ → (ran g × ran g) = ∅)
14	6, 13	eqtr4d 1131	. . 3 ⊢ (g = ∅ → dom g = (ran g × ran g))
15		f1o2 2804	. . . . . 6 ⊢ (g:(x × x)–1-1-onto→x ↔ (g Fn (x × x) ∧ Fun ◡g ∧ ran g = x))
16		fndm 2723	. . . . . . . 8 ⊢ (g Fn (x × x) → dom g = (x × x))
17		xpeq1 2440	. . . . . . . . 9 ⊢ (ran g = x → (ran g × ran g) = (x × ran g))
18		xpeq2 2441	. . . . . . . . 9 ⊢ (ran g = x → (x × ran g) = (x × x))
19	17, 18	eqtr2d 1129	. . . . . . . 8 ⊢ (ran g = x → (x × x) = (ran g × ran g))
20	16, 19	sylan9eq 1144	. . . . . . 7 ⊢ ((g Fn (x × x) ∧ ran g = x) → dom g = (ran g × ran g))
21	20	3adant2 598	. . . . . 6 ⊢ ((g Fn (x × x) ∧ Fun ◡g ∧ ran g = x) → dom g = (ran g × ran g))
22	15, 21	sylbi 174	. . . . 5 ⊢ (g:(x × x)–1-1-onto→x → dom g = (ran g × ran g))
23	22	adantl 305	. . . 4 ⊢ (((ω ≼ x ∧ x ⊆ A) ∧ g:(x × x)–1-1-onto→x) → dom g = (ran g × ran g))
24	23	19.23aiv 952	. . 3 ⊢ (∃x((ω ≼ x ∧ x ⊆ A) ∧ g:(x × x)–1-1-onto→x) → dom g = (ran g × ran g))
25	14, 24	jaoi 275	. 2 ⊢ ((g = ∅ ∨ ∃x((ω ≼ x ∧ x ⊆ A) ∧ g:(x × x)–1-1-onto→x)) → dom g = (ran g × ran g))
26	3, 25	sylbi 174	1 ⊢ (g ∈ H → dom g = (ran g × ran g))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196   ∧ w3a 581  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  ωcom 2372   × cxp 2408  ^◡ccnv 2409  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  –1-1-onto→wf1o 2421   ≼ cdom 3272

This theorem is referenced by:  infxpidmlem5 4937  infxpidmlem7 4939

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-3an 583  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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