Theorem f1imaen2g 6759

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6760 does not need ax-setind 4514.) (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 522	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C e. V )
2		simplr 520	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> B e. V )
3		f1f 5393	. . . . . 6 |- ( F : A -1-1-> B -> F : A --> B )
4		imassrn 4957	. . . . . . 7 |- ( F " C ) C_ ran F
5		frn 5346	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
6	4, 5	sstrid 3153	. . . . . 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 480	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) C_ B )
9	2, 8	ssexd 4122	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) e. _V )
10		f1ores 5447	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
11	10	ad2ant2r 501	. . 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 6721	. . 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 1228	. 2 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C ~~ ( F " C ) )
14	13	ensymd 6749	1 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   ran crn 4605   |` cres 4606   "cima 4607   -->wf 5184   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-er 6501  df-en 6707

This theorem is referenced by:  ssenen  6817  phplem4  6821  phplem4dom  6828  phplem4on  6833  fiintim  6894