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Theorem f1imaen 7533
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
StepHypRef Expression
1 f1imaen.1 . 2  |-  C  e. 
_V
2 f1imaeng 7531 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  _V )  ->  ( F " C
)  ~~  C )
31, 2mp3an3 1313 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C )  ~~  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840   _Vcvv 3056    C_ wss 3411   class class class wbr 4392   "cima 4943   -1-1->wf1 5520    ~~ cen 7469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-er 7266  df-en 7473
This theorem is referenced by:  ssenen  7647  fin4en1  8639  tskinf  9095  tskuni  9109  isercoll  13544  phimullem  14408  odngen  16811  erdsze2lem2  29377
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