Theorem sbthlem9 4455

Description: Lemma for sbth 4457.

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 groups:   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 4453	. . . . . . 7 |- ((Fun f /\ Fun `'g) -> Fun H)
5	1, 2, 3	sbthlem5 4451	. . . . . . . 8 |- ((dom f = A /\ ran g (_ A) -> dom H = A)
6	5	adantrl 394	. . . . . . 7 |- ((dom f = A /\ ((Fun g /\ dom g = B) /\ ran g (_ A)) -> dom H = A)
7	4, 6	anim12i 333	. . . . . 6 |- (((Fun f /\ Fun `'g) /\ (dom f = A /\ ((Fun g /\ dom g = B) /\ ran g (_ A))) -> (Fun H /\ dom H = A))
8	7	an42s 509	. . . . 5 |- (((Fun f /\ dom f = A) /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> (Fun H /\ dom H = A))
9	8	adantlr 393	. . . 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 393	. . 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 4454	. . . . 5 |- ((Fun `'f /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> Fun `'H)
12	11	adantll 392	. . . 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 4452	. . . . . . . 8 |- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> ran H = B)
14		df-rn 3189	. . . . . . . 8 |- ran H = dom `' H
15	13, 14	syl5eqr 1521	. . . . . . 7 |- ((ran f (_ B /\ ((dom g = B /\ ran g (_ A) /\ Fun `'g)) -> dom `' H = B)
16		pm3.27 323	. . . . . . . 8 |- ((Fun g /\ dom g = B) -> dom g = B)
17	16	anim1i 334	. . . . . . 7 |- (((Fun g /\ dom g = B) /\ ran g (_ A) -> (dom g = B /\ ran g (_ A))
18	15, 17	sylanr1 462	. . . . . 6 |- ((ran f (_ B /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> dom `' H = B)
19	18	adantll 392	. . . . 5 |- ((((Fun f /\ dom f = A) /\ ran f (_ B) /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> dom `' H = B)
20	19	adantlr 393	. . . 4 |- (((((Fun f /\ dom f = A) /\ ran f (_ B) /\ Fun `'f) /\ (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g)) -> dom `' H = B)
21	12, 20	jca 288	. . 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))
22	10, 21	jca 288	. 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)))
23		df-f1 3195	. . . 4 |- (f:A-1-1->B <-> (f:A-->B /\ Fun `'f))
24		df-f 3194	. . . . . 6 |- (f:A-->B <-> (f Fn A /\ ran f (_ B))
25		df-fn 3193	. . . . . . 7 |- (f Fn A <-> (Fun f /\ dom f = A))
26	25	anbi1i 481	. . . . . 6 |- ((f Fn A /\ ran f (_ B) <-> ((Fun f /\ dom f = A) /\ ran f (_ B))
27	24, 26	bitr 173	. . . . 5 |- (f:A-->B <-> ((Fun f /\ dom f = A) /\ ran f (_ B))
28	27	anbi1i 481	. . . 4 |- ((f:A-->B /\ Fun `'f) <-> (((Fun f /\ dom f = A) /\ ran f (_ B) /\ Fun `'f))
29	23, 28	bitr 173	. . 3 |- (f:A-1-1->B <-> (((Fun f /\ dom f = A) /\ ran f (_ B) /\ Fun `'f))
30		df-f1 3195	. . . 4 |- (g:B-1-1->A <-> (g:B-->A /\ Fun `'g))
31		df-f 3194	. . . . . 6 |- (g:B-->A <-> (g Fn B /\ ran g (_ A))
32		df-fn 3193	. . . . . . 7 |- (g Fn B <-> (Fun g /\ dom g = B))
33	32	anbi1i 481	. . . . . 6 |- ((g Fn B /\ ran g (_ A) <-> ((Fun g /\ dom g = B) /\ ran g (_ A))
34	31, 33	bitr 173	. . . . 5 |- (g:B-->A <-> ((Fun g /\ dom g = B) /\ ran g (_ A))
35	34	anbi1i 481	. . . 4 |- ((g:B-->A /\ Fun `'g) <-> (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g))
36	30, 35	bitr 173	. . 3 |- (g:B-1-1->A <-> (((Fun g /\ dom g = B) /\ ran g (_ A) /\ Fun `'g))
37	29, 36	anbi12i 482	. 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)))
38		f1o4 3696	. . 3 |- (H:A-1-1-onto->B <-> (H Fn A /\ `'H Fn B))
39		df-fn 3193	. . . 4 |- (H Fn A <-> (Fun H /\ dom H = A))
40		df-fn 3193	. . . 4 |- (`'H Fn B <-> (Fun `'H /\ dom `' H = B))
41	39, 40	anbi12i 482	. . 3 |- ((H Fn A /\ `'H Fn B) <-> ((Fun H /\ dom H = A) /\ (Fun `'H /\ dom `' H = B)))
42	38, 41	bitr 173	. 2 |- (H:A-1-1-onto->B <-> ((Fun H /\ dom H = A) /\ (Fun `'H /\ dom `' H = B)))
43	22, 37, 42	3imtr4 219	1 |- ((f:A-1-1->B /\ g:B-1-1->A) -> H:A-1-1-onto->B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   \ cdif 2044   u. cun 2045   (_ wss 2047  U.cuni 2503  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  -1-1-onto->wf1o 3181

This theorem is referenced by:  sbthlem10 4456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

Copyright terms: Public domain