Theorem domssex2 8120

Description: A corollary of disjenex 8118. If F is an injection from A to B then there is a right inverse g of F from B to a superset of A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
domssex2	|- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> E. g ( g : B -1-1-> _V /\ ( g o. F ) = ( _I |` A ) ) )

Distinct variable groups:   A, g   B, g   g, F

Allowed substitution hints:   V( g)   W( g)

Proof of Theorem domssex2

Step	Hyp	Ref	Expression
1		f1f 6101	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
2		fex2 7121	. . . . 5 |- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1, 2	syl3an1 1359	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F e. _V )
4		f1stres 7190	. . . . . 6 |- ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) --> ( B \ ran F )
5	4	a1i 11	. . . . 5 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) --> ( B \ ran F ) )
6		difexg 4808	. . . . . . 7 |- ( B e. W -> ( B \ ran F ) e. _V )
7	6	3ad2ant3 1084	. . . . . 6 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( B \ ran F ) e. _V )
8		snex 4908	. . . . . 6 |- { ~P U. ran A } e. _V
9		xpexg 6960	. . . . . 6 |- ( ( ( B \ ran F ) e. _V /\ { ~P U. ran A } e. _V ) -> ( ( B \ ran F ) X. { ~P U. ran A } ) e. _V )
10	7, 8, 9	sylancl 694	. . . . 5 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ( B \ ran F ) X. { ~P U. ran A } ) e. _V )
11		fex2 7121	. . . . 5 |- ( ( ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) --> ( B \ ran F ) /\ ( ( B \ ran F ) X. { ~P U. ran A } ) e. _V /\ ( B \ ran F ) e. _V ) -> ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) e. _V )
12	5, 10, 7, 11	syl3anc 1326	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) e. _V )
13		unexg 6959	. . . 4 |- ( ( F e. _V /\ ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) e. _V ) -> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V )
14	3, 12, 13	syl2anc 693	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V )
15		cnvexg 7112	. . 3 |- ( ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V )
16	14, 15	syl 17	. 2 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V )
17		eqid 2622	. . . . . . 7 |- `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
18	17	domss2 8119	. . . . . 6 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) /\ A C_ ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) /\ ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) ) )
19	18	simp1d 1073	. . . . 5 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
20		f1of1 6136	. . . . 5 |- ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
21	19, 20	syl 17	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
22		ssv 3625	. . . 4 |- ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) C_ _V
23		f1ss 6106	. . . 4 |- ( ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) /\ ran `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) C_ _V ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V )
24	21, 22, 23	sylancl 694	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V )
25	18	simp3d 1075	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) )
26	24, 25	jca 554	. 2 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V /\ ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) ) )
27		f1eq1 6096	. . . 4 |- ( g = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> ( g : B -1-1-> _V <-> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V ) )
28		coeq1 5279	. . . . 5 |- ( g = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> ( g o. F ) = ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) )
29	28	eqeq1d 2624	. . . 4 |- ( g = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> ( ( g o. F ) = ( _I |` A ) <-> ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) ) )
30	27, 29	anbi12d 747	. . 3 |- ( g = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> ( ( g : B -1-1-> _V /\ ( g o. F ) = ( _I |` A ) ) <-> ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V /\ ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) ) ) )
31	30	spcegv 3294	. 2 |- ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) e. _V -> ( ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-> _V /\ ( `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) o. F ) = ( _I |` A ) ) -> E. g ( g : B -1-1-> _V /\ ( g o. F ) = ( _I |` A ) ) ) )
32	16, 26, 31	sylc 65	1 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> E. g ( g : B -1-1-> _V /\ ( g o. F ) = ( _I |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   1stc1st 7166

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-en 7956

This theorem is referenced by: (None)