Theorem f1imaen 7036

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)

Hypothesis

Ref	Expression
f1imaen.1	|- C e. _V

Assertion

Ref	Expression
f1imaen	|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C )

Proof of Theorem f1imaen

Step	Hyp	Ref	Expression
1		f1imaen.1	. 2 |- C e. _V
2		f1imaeng 7034	. 2 |- ( ( F : A -1-1-> B /\ C C_ A /\ C e. _V ) -> ( F " C ) ~~ C )
3	1, 2	mp3an3 1363	1 |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   _Vcvv 2815   C_ wss 3213   class class class wbr 4111   "cima 4754   -1-1->wf1 5351   ~~ cen 6975

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-er 6769  df-en 6978

This theorem is referenced by:  ssenen  7107