Theorem sbthlem9 8078

Description: Lemma for sbth 8080. (Contributed by NM, 28-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	|- A e. _V
sbthlem.2	|- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( 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

Allowed substitution hints:   A( f, g)   B( f, g)   D( f, g)   H( f, g)

Proof of Theorem sbthlem9

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . 8 |- A e. _V
2		sbthlem.2	. . . . . . . 8 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
3		sbthlem.3	. . . . . . . 8 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
4	1, 2, 3	sbthlem7 8076	. . . . . . 7 |- ( ( Fun f /\ Fun `' g ) -> Fun H )
5	1, 2, 3	sbthlem5 8074	. . . . . . . 8 |- ( ( dom f = A /\ ran g C_ A ) -> dom H = A )
6	5	adantrl 752	. . . . . . 7 |- ( ( dom f = A /\ ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) ) -> dom H = A )
7	4, 6	anim12i 590	. . . . . 6 |- ( ( ( Fun f /\ Fun `' g ) /\ ( dom f = A /\ ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) ) ) -> ( Fun H /\ dom H = A ) )
8	7	an42s 870	. . . . 5 |- ( ( ( Fun f /\ dom f = A ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
9	8	adantlr 751	. . . 4 |- ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
10	9	adantlr 751	. . 3 |- ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
11	1, 2, 3	sbthlem8 8077	. . . 4 |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
12	11	adantll 750	. . 3 |- ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
13		simpr 477	. . . . . . 7 |- ( ( Fun g /\ dom g = B ) -> dom g = B )
14	13	anim1i 592	. . . . . 6 |- ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> ( dom g = B /\ ran g C_ A ) )
15		df-rn 5125	. . . . . . 7 |- ran H = dom `' H
16	1, 2, 3	sbthlem6 8075	. . . . . . 7 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
17	15, 16	syl5eqr 2670	. . . . . 6 |- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
18	14, 17	sylanr1 684	. . . . 5 |- ( ( ran f C_ B /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
19	18	adantll 750	. . . 4 |- ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
20	19	adantlr 751	. . 3 |- ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
21	10, 12, 20	jca32 558	. 2 |- ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
22		df-f1 5893	. . . 4 |- ( f : A -1-1-> B <-> ( f : A --> B /\ Fun `' f ) )
23		df-f 5892	. . . . . 6 |- ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
24		df-fn 5891	. . . . . . 7 |- ( f Fn A <-> ( Fun f /\ dom f = A ) )
25	24	anbi1i 731	. . . . . 6 |- ( ( f Fn A /\ ran f C_ B ) <-> ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) )
26	23, 25	bitri 264	. . . . 5 |- ( f : A --> B <-> ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) )
27	26	anbi1i 731	. . . 4 |- ( ( f : A --> B /\ Fun `' f ) <-> ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) )
28	22, 27	bitri 264	. . 3 |- ( f : A -1-1-> B <-> ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) )
29		df-f1 5893	. . . 4 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
30		df-f 5892	. . . . . 6 |- ( g : B --> A <-> ( g Fn B /\ ran g C_ A ) )
31		df-fn 5891	. . . . . . 7 |- ( g Fn B <-> ( Fun g /\ dom g = B ) )
32	31	anbi1i 731	. . . . . 6 |- ( ( g Fn B /\ ran g C_ A ) <-> ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) )
33	30, 32	bitri 264	. . . . 5 |- ( g : B --> A <-> ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) )
34	33	anbi1i 731	. . . 4 |- ( ( g : B --> A /\ Fun `' g ) <-> ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) )
35	29, 34	bitri 264	. . 3 |- ( g : B -1-1-> A <-> ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) )
36	28, 35	anbi12i 733	. 2 |- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) <-> ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) )
37		dff1o4 6145	. . 3 |- ( H : A -1-1-onto-> B <-> ( H Fn A /\ `' H Fn B ) )
38		df-fn 5891	. . . 4 |- ( H Fn A <-> ( Fun H /\ dom H = A ) )
39		df-fn 5891	. . . 4 |- ( `' H Fn B <-> ( Fun `' H /\ dom `' H = B ) )
40	38, 39	anbi12i 733	. . 3 |- ( ( H Fn A /\ `' H Fn B ) <-> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
41	37, 40	bitri 264	. 2 |- ( H : A -1-1-onto-> B <-> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
42	21, 36, 41	3imtr4i 281	1 |- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  sbthlem10  8079