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Theorem f1imaen 6364
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 6362 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  _V )  ->  ( F " C
)  ~~  C )
31, 2mp3an3 1258 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C )  ~~  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   _Vcvv 2611    C_ wss 2983   class class class wbr 3806   "cima 4395   -1-1->wf1 4950    ~~ cen 6308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-er 6195  df-en 6311
This theorem is referenced by: (None)
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