Theorem f1imaen2g 6296

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6297 does not need ax-setind 4280.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
f1imaen2g	|- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

Proof of Theorem f1imaen2g

Step	Hyp	Ref	Expression
1		simprr 498	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C e. V )
2		simplr 496	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> B e. V )
3		f1f 5112	. . . . . 6 |- ( F : A -1-1-> B -> F : A --> B )
4		imassrn 4699	. . . . . . 7 |- ( F " C ) C_ ran F
5		frn 5072	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
6	4, 5	syl5ss 3010	. . . . . 6 |- ( F : A --> B -> ( F " C ) C_ B )
7	3, 6	syl 14	. . . . 5 |- ( F : A -1-1-> B -> ( F " C ) C_ B )
8	7	ad2antrr 471	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) C_ B )
9	2, 8	ssexd 3918	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) e. _V )
10		f1ores 5161	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
11	10	ad2ant2r 492	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
12		f1oen2g 6258	. . 3 |- ( ( C e. V /\ ( F " C ) e. _V /\ ( F |` C ) : C -1-1-onto-> ( F " C ) ) -> C ~~ ( F " C ) )
13	1, 9, 11, 12	syl3anc 1169	. 2 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C ~~ ( F " C ) )
14	13	ensymd 6286	1 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   _Vcvv 2601   C_ wss 2973   class class class wbr 3785   ran crn 4364   |` cres 4365   "cima 4366   -->wf 4918   -1-1->wf1 4919   -1-1-onto->wf1o 4921   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-er 6129  df-en 6245

This theorem is referenced by:  phplem4  6341  phplem4dom  6348  phplem4on  6353