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Theorem f1imaen 6939
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 6937 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  _V )  ->  ( F " C
)  ~~  C )
31, 2mp3an3 1266 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C )  ~~  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   "cima 4708   -1-1->wf1 5268    ~~ cen 6876
This theorem is referenced by:  ssenen  7051  fin4en1  7951  tskinf  8407  tskuni  8421  isercoll  12157  phimullem  12863  odngen  14904  erdsze2lem2  23750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880
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