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Theorem f1imaen2g 6655
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6656 does not need ax-setind 4422.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
f1imaen2g  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  ~~  C )

Proof of Theorem f1imaen2g
StepHypRef Expression
1 simprr 506 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  e.  V )
2 simplr 504 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  B  e.  V )
3 f1f 5298 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
4 imassrn 4862 . . . . . . 7  |-  ( F
" C )  C_  ran  F
5 frn 5251 . . . . . . 7  |-  ( F : A --> B  ->  ran  F  C_  B )
64, 5sstrid 3078 . . . . . 6  |-  ( F : A --> B  -> 
( F " C
)  C_  B )
73, 6syl 14 . . . . 5  |-  ( F : A -1-1-> B  -> 
( F " C
)  C_  B )
87ad2antrr 479 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  C_  B )
92, 8ssexd 4038 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  e.  _V )
10 f1ores 5350 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
1110ad2ant2r 500 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F  |`  C ) : C -1-1-onto-> ( F " C
) )
12 f1oen2g 6617 . . 3  |-  ( ( C  e.  V  /\  ( F " C )  e.  _V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
131, 9, 11, 12syl3anc 1201 . 2  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  ~~  ( F " C ) )
1413ensymd 6645 1  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  ~~  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   _Vcvv 2660    C_ wss 3041   class class class wbr 3899   ran crn 4510    |` cres 4511   "cima 4512   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-er 6397  df-en 6603
This theorem is referenced by:  ssenen  6713  phplem4  6717  phplem4dom  6724  phplem4on  6729  fiintim  6785
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