Theorem f1imaen2g 6966

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6967 does not need ax-setind 4635.) (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 533	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C e. V )
2		simplr 529	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> B e. V )
3		f1f 5542	. . . . . 6 |- ( F : A -1-1-> B -> F : A --> B )
4		imassrn 5087	. . . . . . 7 |- ( F " C ) C_ ran F
5		frn 5491	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
6	4, 5	sstrid 3238	. . . . . 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 488	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) C_ B )
9	2, 8	ssexd 4229	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) e. _V )
10		f1ores 5598	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
11	10	ad2ant2r 509	. . 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 6927	. . 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 1273	. 2 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C ~~ ( F " C ) )
14	13	ensymd 6956	1 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   ran crn 4726   |` cres 4727   "cima 4728   -->wf 5322   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ~~ cen 6906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-er 6701  df-en 6909

This theorem is referenced by:  ssenen  7036  phplem4  7040  phplem4dom  7047  phplem4on  7053  fiintim  7122