Theorem f1imaen2g 8017

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 8018 does not need ax-reg 8497.) (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 796	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C e. V )
2		simplr 792	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> B e. V )
3		f1f 6101	. . . . . 6 |- ( F : A -1-1-> B -> F : A --> B )
4		imassrn 5477	. . . . . . 7 |- ( F " C ) C_ ran F
5		frn 6053	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
6	4, 5	syl5ss 3614	. . . . . 6 |- ( F : A --> B -> ( F " C ) C_ B )
7	3, 6	syl 17	. . . . 5 |- ( F : A -1-1-> B -> ( F " C ) C_ B )
8	7	ad2antrr 762	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) C_ B )
9	2, 8	ssexd 4805	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) e. _V )
10		f1ores 6151	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
11	10	ad2ant2r 783	. . 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 7972	. . 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 1326	. 2 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C ~~ ( F " C ) )
14	13	ensymd 8007	1 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ~~ 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-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-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

This theorem is referenced by:  ssenen  8134  phplem4  8142  fiint  8237  unxpwdom2  8493  znunithash  19913