Theorem f1imaen2g 7080

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7081 does not need ax-setind 4684.) (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 537	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C e. V )
2		simplr 533	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> B e. V )
3		f1f 5598	. . . . . 6 |- ( F : A -1-1-> B -> F : A --> B )
4		imassrn 5137	. . . . . . 7 |- ( F " C ) C_ ran F
5		frn 5542	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
6	4, 5	sstrid 3259	. . . . . 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 492	. . . 4 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) C_ B )
9	2, 8	ssexd 4273	. . 3 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) e. _V )
10		f1ores 5654	. . . 4 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
11	10	ad2ant2r 513	. . 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 7041	. . 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 1278	. 2 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> C ~~ ( F " C ) )
14	13	ensymd 7070	1 |- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4130   ran crn 4775   |` cres 4776   "cima 4777   -->wf 5373   -1-1->wf1 5374   -1-1-onto->wf1o 5376   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-er 6807  df-en 7023

This theorem is used by:  ssenen  7152  phplem4  7156  phplem4dom  7163  phplem4on  7169  fiintim  7238