Theorem sbthlem10 3358

Description: Lemma for Schroeder-Bernstein Theorem.

Hypotheses

Ref	Expression
sbthlem.1	⊢ A ∈ V
sbthlem.2	⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
sbthlem.3	⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
sbthlem.4	⊢ B ∈ V

Assertion

Ref	Expression
sbthlem10	⊢ ((A ≼ B ∧ B ≼ A) → A ≈ B)

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

Proof of Theorem sbthlem10

Step	Hyp	Ref	Expression
1		sbthlem.4	. . . . 5 ⊢ B ∈ V
2	1	brdom 3283	. . . 4 ⊢ (A ≼ B ↔ ∃f f:A–1-1→B)
3		sbthlem.1	. . . . 5 ⊢ A ∈ V
4	3	brdom 3283	. . . 4 ⊢ (B ≼ A ↔ ∃g g:B–1-1→A)
5	2, 4	anbi12i 369	. . 3 ⊢ ((A ≼ B ∧ B ≼ A) ↔ (∃f f:A–1-1→B ∧ ∃g g:B–1-1→A))
6		eeanv 980	. . 3 ⊢ (∃f∃g(f:A–1-1→B ∧ g:B–1-1→A) ↔ (∃f f:A–1-1→B ∧ ∃g g:B–1-1→A))
7	5, 6	bitr4 154	. 2 ⊢ ((A ≼ B ∧ B ≼ A) ↔ ∃f∃g(f:A–1-1→B ∧ g:B–1-1→A))
8		sbthlem.2	. . . . 5 ⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
9		sbthlem.3	. . . . 5 ⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
10	3, 8, 9	sbthlem9 3357	. . . 4 ⊢ ((f:A–1-1→B ∧ g:B–1-1→A) → H:A–1-1-onto→B)
11	3	f1oen 3301	. . . 4 ⊢ (H:A–1-1-onto→B → A ≈ B)
12	10, 11	syl 12	. . 3 ⊢ ((f:A–1-1→B ∧ g:B–1-1→A) → A ≈ B)
13	12	19.23aivv 953	. 2 ⊢ (∃f∃g(f:A–1-1→B ∧ g:B–1-1→A) → A ≈ B)
14	7, 13	sylbi 174	1 ⊢ ((A ≼ B ∧ B ≼ A) → A ≈ B)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487  ∪cuni 1919   class class class wbr 2054  ^◡ccnv 2409   ↾ cres 2412   “ cima 2413  –1-1→wf1 2419  –1-1-onto→wf1o 2421   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  sbth 3359

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-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-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-uni 1920  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-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274  df-dom 3275

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