Theorem wdomima2g 8491

Description: A set is weakly dominant over its image under any function. This version of wdomimag 8492 is stated so as to avoid ax-rep 4771. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
wdomima2g	|- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F " A ) ~<_* A )

Proof of Theorem wdomima2g

Step	Hyp	Ref	Expression
1		df-ima 5127	. 2 |- ( F " A ) = ran ( F |` A )
2		funres 5929	. . . . . . . 8 |- ( Fun F -> Fun ( F |` A ) )
3		funforn 6122	. . . . . . . 8 |- ( Fun ( F |` A ) <-> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
4	2, 3	sylib 208	. . . . . . 7 |- ( Fun F -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
5	4	3ad2ant1 1082	. . . . . 6 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
6		fof 6115	. . . . . 6 |- ( ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) -> ( F |` A ) : dom ( F |` A ) --> ran ( F |` A ) )
7	5, 6	syl 17	. . . . 5 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F |` A ) : dom ( F |` A ) --> ran ( F |` A ) )
8		dmres 5419	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
9		inss1 3833	. . . . . . 7 |- ( A i^i dom F ) C_ A
10	8, 9	eqsstri 3635	. . . . . 6 |- dom ( F |` A ) C_ A
11		simp2 1062	. . . . . 6 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> A e. V )
12		ssexg 4804	. . . . . 6 |- ( ( dom ( F |` A ) C_ A /\ A e. V ) -> dom ( F |` A ) e. _V )
13	10, 11, 12	sylancr 695	. . . . 5 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> dom ( F |` A ) e. _V )
14		simp3 1063	. . . . . 6 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F " A ) e. W )
15	1, 14	syl5eqelr 2706	. . . . 5 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ran ( F |` A ) e. W )
16		fex2 7121	. . . . 5 |- ( ( ( F |` A ) : dom ( F |` A ) --> ran ( F |` A ) /\ dom ( F |` A ) e. _V /\ ran ( F |` A ) e. W ) -> ( F |` A ) e. _V )
17	7, 13, 15, 16	syl3anc 1326	. . . 4 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F |` A ) e. _V )
18		fowdom 8476	. . . 4 |- ( ( ( F |` A ) e. _V /\ ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) ) -> ran ( F |` A ) ~<_* dom ( F |` A ) )
19	17, 5, 18	syl2anc 693	. . 3 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ran ( F |` A ) ~<_* dom ( F |` A ) )
20		ssdomg 8001	. . . . . 6 |- ( A e. V -> ( dom ( F |` A ) C_ A -> dom ( F |` A ) ~<_ A ) )
21	10, 20	mpi 20	. . . . 5 |- ( A e. V -> dom ( F |` A ) ~<_ A )
22		domwdom 8479	. . . . 5 |- ( dom ( F |` A ) ~<_ A -> dom ( F |` A ) ~<_* A )
23	21, 22	syl 17	. . . 4 |- ( A e. V -> dom ( F |` A ) ~<_* A )
24	23	3ad2ant2 1083	. . 3 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> dom ( F |` A ) ~<_* A )
25		wdomtr 8480	. . 3 |- ( ( ran ( F |` A ) ~<_* dom ( F |` A ) /\ dom ( F |` A ) ~<_* A ) -> ran ( F |` A ) ~<_* A )
26	19, 24, 25	syl2anc 693	. 2 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ran ( F |` A ) ~<_* A )
27	1, 26	syl5eqbr 4688	1 |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) -> ( F " A ) ~<_* A )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -onto->wfo 5886   ~<_ cdom 7953   ~<_^* cwdom 8462

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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464

This theorem is referenced by:  wdomimag  8492  unxpwdom2  8493