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Theorem f1imaen2g 7105
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7106 does not need ax-reg 7494.) (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 734 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  e.  V )
2 simplr 732 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  B  e.  V )
3 f1f 5580 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
4 imassrn 5157 . . . . . . 7  |-  ( F
" C )  C_  ran  F
5 frn 5538 . . . . . . 7  |-  ( F : A --> B  ->  ran  F  C_  B )
64, 5syl5ss 3303 . . . . . 6  |-  ( F : A --> B  -> 
( F " C
)  C_  B )
73, 6syl 16 . . . . 5  |-  ( F : A -1-1-> B  -> 
( F " C
)  C_  B )
87ad2antrr 707 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  C_  B )
92, 8ssexd 4292 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  e.  _V )
10 f1ores 5630 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
1110ad2ant2r 728 . . 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 7061 . . 3  |-  ( ( C  e.  V  /\  ( F " C )  e.  _V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
131, 9, 11, 12syl3anc 1184 . 2  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  ~~  ( F " C ) )
1413ensymd 7095 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 359    e. wcel 1717   _Vcvv 2900    C_ wss 3264   class class class wbr 4154   ran crn 4820    |` cres 4821   "cima 4822   -->wf 5391   -1-1->wf1 5392   -1-1-onto->wf1o 5394    ~~ cen 7043
This theorem is referenced by:  ssenen  7218  phplem4  7226  fiint  7320  unxpwdom2  7490  znunithash  16769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-er 6842  df-en 7047
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