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Theorem f1imaeng 6272
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
f1imaeng  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )

Proof of Theorem f1imaeng
StepHypRef Expression
1 f1ores 5141 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
2 f1oeng 6237 . . . 4  |-  ( ( C  e.  V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
32ancoms 255 . . 3  |-  ( ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  /\  C  e.  V )  ->  C  ~~  ( F " C
) )
41, 3stoic3 1320 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  C  ~~  ( F
" C ) )
54ensymd 6263 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 885    e. wcel 1393    C_ wss 2917   class class class wbr 3764    |` cres 4347   "cima 4348   -1-1->wf1 4899   -1-1-onto->wf1o 4901    ~~ cen 6219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222
This theorem is referenced by:  f1imaen  6274
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