Theorem sbthlem9 3357

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)))

Assertion

Ref	Expression
sbthlem9	⊢ ((f:A–1-1→B ∧ g:B–1-1→A) → H:A–1-1-onto→B)

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

Proof of Theorem sbthlem9

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . 8 ⊢ A ∈ V
2		sbthlem.2	. . . . . . . 8 ⊢ D = {x∣(x ⊆ A ∧ (g “ (B ∖ (f “ x))) ⊆ (A ∖ x))}
3		sbthlem.3	. . . . . . . 8 ⊢ H = ((f ↾ ∪D) ∪ (◡g ↾ (A ∖ ∪D)))
4	1, 2, 3	sbthlem7 3355	. . . . . . 7 ⊢ ((Fun f ∧ Fun ◡g) → Fun H)
5	1, 2, 3	sbthlem5 3353	. . . . . . . 8 ⊢ ((dom f = A ∧ ran g ⊆ A) → dom H = A)
6	5	adantrl 311	. . . . . . 7 ⊢ ((dom f = A ∧ ((Fun g ∧ dom g = B) ∧ ran g ⊆ A)) → dom H = A)
7	4, 6	anim12i 268	. . . . . 6 ⊢ (((Fun f ∧ Fun ◡g) ∧ (dom f = A ∧ ((Fun g ∧ dom g = B) ∧ ran g ⊆ A))) → (Fun H ∧ dom H = A))
8	7	an42s 391	. . . . 5 ⊢ (((Fun f ∧ dom f = A) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (Fun H ∧ dom H = A))
9	8	adantlr 310	. . . 4 ⊢ ((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (Fun H ∧ dom H = A))
10	9	adantlr 310	. . 3 ⊢ (((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (Fun H ∧ dom H = A))
11	1, 2, 3	sbthlem8 3356	. . . . 5 ⊢ ((Fun ◡f ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → Fun ◡H)
12	11	adantll 309	. . . 4 ⊢ (((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → Fun ◡H)
13	1, 2, 3	sbthlem6 3354	. . . . . . . 8 ⊢ ((ran f ⊆ B ∧ ((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g)) → ran H = B)
14		df-rn 2429	. . . . . . . 8 ⊢ ran H = dom ◡H
15	13, 14	syl5eqr 1138	. . . . . . 7 ⊢ ((ran f ⊆ B ∧ ((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g)) → dom ◡H = B)
16		pm3.27 260	. . . . . . . . 9 ⊢ ((Fun g ∧ dom g = B) → dom g = B)
17	16	anim1i 269	. . . . . . . 8 ⊢ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) → (dom g = B ∧ ran g ⊆ A))
18	17	anim1i 269	. . . . . . 7 ⊢ ((((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g) → ((dom g = B ∧ ran g ⊆ A) ∧ Fun ◡g))
19	15, 18	sylan2 346	. . . . . 6 ⊢ ((ran f ⊆ B ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → dom ◡H = B)
20	19	adantll 309	. . . . 5 ⊢ ((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → dom ◡H = B)
21	20	adantlr 310	. . . 4 ⊢ (((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → dom ◡H = B)
22	12, 21	jca 236	. . 3 ⊢ (((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → (Fun ◡H ∧ dom ◡H = B))
23	10, 22	jca 236	. 2 ⊢ (((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)) → ((Fun H ∧ dom H = A) ∧ (Fun ◡H ∧ dom ◡H = B)))
24		df-f1 2435	. . . 4 ⊢ (f:A–1-1→B ↔ (f:A–→B ∧ Fun ◡f))
25		df-f 2434	. . . . . 6 ⊢ (f:A–→B ↔ (f Fn A ∧ ran f ⊆ B))
26		df-fn 2433	. . . . . . 7 ⊢ (f Fn A ↔ (Fun f ∧ dom f = A))
27	26	anbi1i 368	. . . . . 6 ⊢ ((f Fn A ∧ ran f ⊆ B) ↔ ((Fun f ∧ dom f = A) ∧ ran f ⊆ B))
28	25, 27	bitr 151	. . . . 5 ⊢ (f:A–→B ↔ ((Fun f ∧ dom f = A) ∧ ran f ⊆ B))
29	28	anbi1i 368	. . . 4 ⊢ ((f:A–→B ∧ Fun ◡f) ↔ (((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f))
30	24, 29	bitr 151	. . 3 ⊢ (f:A–1-1→B ↔ (((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f))
31		df-f1 2435	. . . 4 ⊢ (g:B–1-1→A ↔ (g:B–→A ∧ Fun ◡g))
32		df-f 2434	. . . . . 6 ⊢ (g:B–→A ↔ (g Fn B ∧ ran g ⊆ A))
33		df-fn 2433	. . . . . . 7 ⊢ (g Fn B ↔ (Fun g ∧ dom g = B))
34	33	anbi1i 368	. . . . . 6 ⊢ ((g Fn B ∧ ran g ⊆ A) ↔ ((Fun g ∧ dom g = B) ∧ ran g ⊆ A))
35	32, 34	bitr 151	. . . . 5 ⊢ (g:B–→A ↔ ((Fun g ∧ dom g = B) ∧ ran g ⊆ A))
36	35	anbi1i 368	. . . 4 ⊢ ((g:B–→A ∧ Fun ◡g) ↔ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g))
37	31, 36	bitr 151	. . 3 ⊢ (g:B–1-1→A ↔ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g))
38	30, 37	anbi12i 369	. 2 ⊢ ((f:A–1-1→B ∧ g:B–1-1→A) ↔ ((((Fun f ∧ dom f = A) ∧ ran f ⊆ B) ∧ Fun ◡f) ∧ (((Fun g ∧ dom g = B) ∧ ran g ⊆ A) ∧ Fun ◡g)))
39		f1o4 2807	. . 3 ⊢ (H:A–1-1-onto→B ↔ (H Fn A ∧ ◡H Fn B))
40		df-fn 2433	. . . 4 ⊢ (H Fn A ↔ (Fun H ∧ dom H = A))
41		df-fn 2433	. . . 4 ⊢ (◡H Fn B ↔ (Fun ◡H ∧ dom ◡H = B))
42	40, 41	anbi12i 369	. . 3 ⊢ ((H Fn A ∧ ◡H Fn B) ↔ ((Fun H ∧ dom H = A) ∧ (Fun ◡H ∧ dom ◡H = B)))
43	39, 42	bitr 151	. 2 ⊢ (H:A–1-1-onto→B ↔ ((Fun H ∧ dom H = A) ∧ (Fun ◡H ∧ dom ◡H = B)))
44	23, 38, 43	3imtr4 192	1 ⊢ ((f:A–1-1→B ∧ g:B–1-1→A) → H:A–1-1-onto→B)

Syntax hints:   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487  ∪cuni 1919  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417  –→wf 2418  –1-1→wf1 2419  –1-1-onto→wf1o 2421

This theorem is referenced by:  sbthlem10 3358

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-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

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