Theorem domss2 8119

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

Hypothesis

Ref	Expression
domss2.1	|- G = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )

Assertion

Ref	Expression
domss2	|- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( G : B -1-1-onto-> ran G /\ A C_ ran G /\ ( G o. F ) = ( _I |` A ) ) )

Proof of Theorem domss2

Step	Hyp	Ref	Expression
1		f1f1orn 6148	. . . . . . . 8 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
2	1	3ad2ant1 1082	. . . . . . 7 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F : A -1-1-onto-> ran F )
3		simp2 1062	. . . . . . . . . 10 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> A e. V )
4		rnexg 7098	. . . . . . . . . 10 |- ( A e. V -> ran A e. _V )
5	3, 4	syl 17	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ran A e. _V )
6		uniexg 6955	. . . . . . . . 9 |- ( ran A e. _V -> U. ran A e. _V )
7		pwexg 4850	. . . . . . . . 9 |- ( U. ran A e. _V -> ~P U. ran A e. _V )
8	5, 6, 7	3syl 18	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ~P U. ran A e. _V )
9		1stconst 7265	. . . . . . . 8 |- ( ~P U. ran A e. _V -> ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) -1-1-onto-> ( B \ ran F ) )
10	8, 9	syl 17	. . . . . . 7 |- ( ( 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 } ) -1-1-onto-> ( B \ ran F ) )
11		difexg 4808	. . . . . . . . . 10 |- ( B e. W -> ( B \ ran F ) e. _V )
12	11	3ad2ant3 1084	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( B \ ran F ) e. _V )
13		disjen 8117	. . . . . . . . 9 |- ( ( A e. V /\ ( B \ ran F ) e. _V ) -> ( ( A i^i ( ( B \ ran F ) X. { ~P U. ran A } ) ) = (/) /\ ( ( B \ ran F ) X. { ~P U. ran A } ) ~~ ( B \ ran F ) ) )
14	3, 12, 13	syl2anc 693	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ( A i^i ( ( B \ ran F ) X. { ~P U. ran A } ) ) = (/) /\ ( ( B \ ran F ) X. { ~P U. ran A } ) ~~ ( B \ ran F ) ) )
15	14	simpld 475	. . . . . . 7 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( A i^i ( ( B \ ran F ) X. { ~P U. ran A } ) ) = (/) )
16		disjdif 4040	. . . . . . . 8 |- ( ran F i^i ( B \ ran F ) ) = (/)
17	16	a1i 11	. . . . . . 7 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ran F i^i ( B \ ran F ) ) = (/) )
18		f1oun 6156	. . . . . . 7 |- ( ( ( F : A -1-1-onto-> ran F /\ ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) -1-1-onto-> ( B \ ran F ) ) /\ ( ( A i^i ( ( B \ ran F ) X. { ~P U. ran A } ) ) = (/) /\ ( ran F i^i ( B \ ran F ) ) = (/) ) ) -> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> ( ran F u. ( B \ ran F ) ) )
19	2, 10, 15, 17, 18	syl22anc 1327	. . . . . 6 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> ( ran F u. ( B \ ran F ) ) )
20		undif2 4044	. . . . . . . 8 |- ( ran F u. ( B \ ran F ) ) = ( ran F u. B )
21		f1f 6101	. . . . . . . . . . 11 |- ( F : A -1-1-> B -> F : A --> B )
22	21	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F : A --> B )
23		frn 6053	. . . . . . . . . 10 |- ( F : A --> B -> ran F C_ B )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ran F C_ B )
25		ssequn1 3783	. . . . . . . . 9 |- ( ran F C_ B <-> ( ran F u. B ) = B )
26	24, 25	sylib 208	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ran F u. B ) = B )
27	20, 26	syl5eq 2668	. . . . . . 7 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ran F u. ( B \ ran F ) ) = B )
28		f1oeq3 6129	. . . . . . 7 |- ( ( ran F u. ( B \ ran F ) ) = B -> ( ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> ( ran F u. ( B \ ran F ) ) <-> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> B ) )
29	27, 28	syl 17	. . . . . 6 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> ( ran F u. ( B \ ran F ) ) <-> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> B ) )
30	19, 29	mpbid 222	. . . . 5 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> B )
31		f1ocnv 6149	. . . . 5 |- ( ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -1-1-onto-> B -> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
32	30, 31	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-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
33		domss2.1	. . . . 5 |- G = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
34		f1oeq1 6127	. . . . 5 |- ( G = `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) -> ( G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) <-> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
35	33, 34	ax-mp 5	. . . 4 |- ( G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) <-> `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
36	32, 35	sylibr 224	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
37		f1ofo 6144	. . . . 5 |- ( G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -> G : B -onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
38		forn 6118	. . . . 5 |- ( G : B -onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -> ran G = ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
39	36, 37, 38	3syl 18	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ran G = ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
40		f1oeq3 6129	. . . 4 |- ( ran G = ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) -> ( G : B -1-1-onto-> ran G <-> G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
41	39, 40	syl 17	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( G : B -1-1-onto-> ran G <-> G : B -1-1-onto-> ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) )
42	36, 41	mpbird 247	. 2 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> G : B -1-1-onto-> ran G )
43		ssun1 3776	. . 3 |- A C_ ( A u. ( ( B \ ran F ) X. { ~P U. ran A } ) )
44	43, 39	syl5sseqr 3654	. 2 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> A C_ ran G )
45		ssid 3624	. . . 4 |- ran F C_ ran F
46		cores 5638	. . . 4 |- ( ran F C_ ran F -> ( ( G |` ran F ) o. F ) = ( G o. F ) )
47	45, 46	ax-mp 5	. . 3 |- ( ( G |` ran F ) o. F ) = ( G o. F )
48		dmres 5419	. . . . . . . . 9 |- dom ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = ( ran F i^i dom `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
49		f1ocnv 6149	. . . . . . . . . . . 12 |- ( ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( ( B \ ran F ) X. { ~P U. ran A } ) -1-1-onto-> ( B \ ran F ) -> `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( B \ ran F ) -1-1-onto-> ( ( B \ ran F ) X. { ~P U. ran A } ) )
50		f1odm 6141	. . . . . . . . . . . 12 |- ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) : ( B \ ran F ) -1-1-onto-> ( ( B \ ran F ) X. { ~P U. ran A } ) -> dom `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) = ( B \ ran F ) )
51	10, 49, 50	3syl 18	. . . . . . . . . . 11 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> dom `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) = ( B \ ran F ) )
52	51	ineq2d 3814	. . . . . . . . . 10 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ran F i^i dom `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) = ( ran F i^i ( B \ ran F ) ) )
53	52, 16	syl6eq 2672	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ran F i^i dom `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) = (/) )
54	48, 53	syl5eq 2668	. . . . . . . 8 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> dom ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) )
55		relres 5426	. . . . . . . . 9 |- Rel ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F )
56		reldm0 5343	. . . . . . . . 9 |- ( Rel ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) -> ( ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) <-> dom ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) ) )
57	55, 56	ax-mp 5	. . . . . . . 8 |- ( ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) <-> dom ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) )
58	54, 57	sylibr 224	. . . . . . 7 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) = (/) )
59	58	uneq2d 3767	. . . . . 6 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' F u. ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) ) = ( `' F u. (/) ) )
60		cnvun 5538	. . . . . . . . 9 |- `' ( F u. ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) = ( `' F u. `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
61	33, 60	eqtri 2644	. . . . . . . 8 |- G = ( `' F u. `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) )
62	61	reseq1i 5392	. . . . . . 7 |- ( G |` ran F ) = ( ( `' F u. `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) |` ran F )
63		resundir 5411	. . . . . . 7 |- ( ( `' F u. `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) |` ran F ) = ( ( `' F |` ran F ) u. ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) )
64		df-rn 5125	. . . . . . . . . 10 |- ran F = dom `' F
65	64	reseq2i 5393	. . . . . . . . 9 |- ( `' F |` ran F ) = ( `' F |` dom `' F )
66		relcnv 5503	. . . . . . . . . 10 |- Rel `' F
67		resdm 5441	. . . . . . . . . 10 |- ( Rel `' F -> ( `' F |` dom `' F ) = `' F )
68	66, 67	ax-mp 5	. . . . . . . . 9 |- ( `' F |` dom `' F ) = `' F
69	65, 68	eqtri 2644	. . . . . . . 8 |- ( `' F |` ran F ) = `' F
70	69	uneq1i 3763	. . . . . . 7 |- ( ( `' F |` ran F ) u. ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) ) = ( `' F u. ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) )
71	62, 63, 70	3eqtrri 2649	. . . . . 6 |- ( `' F u. ( `' ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) |` ran F ) ) = ( G |` ran F )
72		un0 3967	. . . . . 6 |- ( `' F u. (/) ) = `' F
73	59, 71, 72	3eqtr3g 2679	. . . . 5 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( G |` ran F ) = `' F )
74	73	coeq1d 5283	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ( G |` ran F ) o. F ) = ( `' F o. F ) )
75		f1cocnv1 6166	. . . . 5 |- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) )
76	75	3ad2ant1 1082	. . . 4 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( `' F o. F ) = ( _I |` A ) )
77	74, 76	eqtrd 2656	. . 3 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( ( G |` ran F ) o. F ) = ( _I |` A ) )
78	47, 77	syl5eqr 2670	. 2 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( G o. F ) = ( _I |` A ) )
79	42, 44, 78	3jca 1242	1 |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> ( G : B -1-1-onto-> ran G /\ A C_ ran G /\ ( G o. F ) = ( _I |` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   1stc1st 7166   ~~ cen 7952

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:  domssex2  8120  domssex  8121