Theorem sbthlem9 3357

Description: Lemma for Schroeder-Bernstein Theorem.

Hypotheses

Ref	Expression
sbthlem.1	|- A e. V
sbthlem.2	|- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
sbthlem.3	|- H = ((f |` U.D) u. (`'g |` (A \ U.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 e. V
2		sbthlem.2	. . . . . . . 8 |- D = {x | (x (_ A /\ (g"(B \ (f"x))) (_ (A \ x))}
3		sbthlem.3	. . . . . . . 8 |- H = ((f |` U.D) u. (`'g |` (A \ U.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   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   (_ wss 1487  U.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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