Theorem domssex 8121

Description: Weakening of domssex 8121 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
domssex	|- ( A ~<_ B -> E. x ( A C_ x /\ B ~~ x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem domssex

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 7966	. 2 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2		reldom 7961	. . 3 |- Rel ~<_
3	2	brrelex2i 5159	. 2 |- ( A ~<_ B -> B e. _V )
4		vex 3203	. . . . . . . 8 |- f e. _V
5		f1stres 7190	. . . . . . . . . 10 |- ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) : ( ( B \ ran f ) X. { ~P U. ran A } ) --> ( B \ ran f )
6	5	a1i 11	. . . . . . . . 9 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) : ( ( B \ ran f ) X. { ~P U. ran A } ) --> ( B \ ran f ) )
7		difexg 4808	. . . . . . . . . . 11 |- ( B e. _V -> ( B \ ran f ) e. _V )
8	7	adantl 482	. . . . . . . . . 10 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( B \ ran f ) e. _V )
9		snex 4908	. . . . . . . . . 10 |- { ~P U. ran A } e. _V
10		xpexg 6960	. . . . . . . . . 10 |- ( ( ( B \ ran f ) e. _V /\ { ~P U. ran A } e. _V ) -> ( ( B \ ran f ) X. { ~P U. ran A } ) e. _V )
11	8, 9, 10	sylancl 694	. . . . . . . . 9 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( ( B \ ran f ) X. { ~P U. ran A } ) e. _V )
12		fex2 7121	. . . . . . . . 9 |- ( ( ( 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 )
13	6, 11, 8, 12	syl3anc 1326	. . . . . . . 8 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) e. _V )
14		unexg 6959	. . . . . . . 8 |- ( ( 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 )
15	4, 13, 14	sylancr 695	. . . . . . 7 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V )
16		cnvexg 7112	. . . . . . 7 |- ( ( 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 )
17	15, 16	syl 17	. . . . . 6 |- ( ( f : A -1-1-> B /\ B e. _V ) -> `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V )
18		rnexg 7098	. . . . . 6 |- ( `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V -> ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V )
19	17, 18	syl 17	. . . . 5 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V )
20		simpl 473	. . . . . . . 8 |- ( ( f : A -1-1-> B /\ B e. _V ) -> f : A -1-1-> B )
21		f1dm 6105	. . . . . . . . . 10 |- ( f : A -1-1-> B -> dom f = A )
22	4	dmex 7099	. . . . . . . . . 10 |- dom f e. _V
23	21, 22	syl6eqelr 2710	. . . . . . . . 9 |- ( f : A -1-1-> B -> A e. _V )
24	23	adantr 481	. . . . . . . 8 |- ( ( f : A -1-1-> B /\ B e. _V ) -> A e. _V )
25		simpr 477	. . . . . . . 8 |- ( ( f : A -1-1-> B /\ B e. _V ) -> B e. _V )
26		eqid 2622	. . . . . . . . 9 |- `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) = `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) )
27	26	domss2 8119	. . . . . . . 8 |- ( ( f : A -1-1-> B /\ A e. _V /\ B e. _V ) -> ( `' ( 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 ) ) )
28	20, 24, 25, 27	syl3anc 1326	. . . . . . 7 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( `' ( 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 ) ) )
29	28	simp2d 1074	. . . . . 6 |- ( ( f : A -1-1-> B /\ B e. _V ) -> A C_ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) )
30	28	simp1d 1073	. . . . . . 7 |- ( ( f : A -1-1-> B /\ B e. _V ) -> `' ( 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 } ) ) ) )
31		f1oen3g 7971	. . . . . . 7 |- ( ( `' ( 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-onto-> ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) -> B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) )
32	17, 30, 31	syl2anc 693	. . . . . 6 |- ( ( f : A -1-1-> B /\ B e. _V ) -> B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) )
33	29, 32	jca 554	. . . . 5 |- ( ( f : A -1-1-> B /\ B e. _V ) -> ( A C_ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) /\ B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) )
34		sseq2 3627	. . . . . . 7 |- ( x = ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) -> ( A C_ x <-> A C_ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) )
35		breq2 4657	. . . . . . 7 |- ( x = ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) -> ( B ~~ x <-> B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) )
36	34, 35	anbi12d 747	. . . . . 6 |- ( x = ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) -> ( ( A C_ x /\ B ~~ x ) <-> ( A C_ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) /\ B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) ) )
37	36	spcegv 3294	. . . . 5 |- ( ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) e. _V -> ( ( A C_ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) /\ B ~~ ran `' ( f u. ( 1st |` ( ( B \ ran f ) X. { ~P U. ran A } ) ) ) ) -> E. x ( A C_ x /\ B ~~ x ) ) )
38	19, 33, 37	sylc 65	. . . 4 |- ( ( f : A -1-1-> B /\ B e. _V ) -> E. x ( A C_ x /\ B ~~ x ) )
39	38	ex 450	. . 3 |- ( f : A -1-1-> B -> ( B e. _V -> E. x ( A C_ x /\ B ~~ x ) ) )
40	39	exlimiv 1858	. 2 |- ( E. f f : A -1-1-> B -> ( B e. _V -> E. x ( A C_ x /\ B ~~ x ) ) )
41	1, 3, 40	sylc 65	1 |- ( A ~<_ B -> E. x ( A C_ x /\ B ~~ x ) )

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   class class class wbr 4653   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   1stc1st 7166   ~~ cen 7952   ~<_ cdom 7953

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  df-dom 7957

This theorem is referenced by: (None)