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Theorem f1imaen2g 7616
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7617 does not need ax-reg 8054.) (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 760 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  e.  V )
2 simplr 756 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  B  e.  V )
3 f1f 5766 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
4 imassrn 5170 . . . . . . 7  |-  ( F
" C )  C_  ran  F
5 frn 5722 . . . . . . 7  |-  ( F : A --> B  ->  ran  F  C_  B )
64, 5syl5ss 3455 . . . . . 6  |-  ( F : A --> B  -> 
( F " C
)  C_  B )
73, 6syl 17 . . . . 5  |-  ( F : A -1-1-> B  -> 
( F " C
)  C_  B )
87ad2antrr 726 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  C_  B )
92, 8ssexd 4543 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  e.  _V )
10 f1ores 5815 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
1110ad2ant2r 747 . . 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 7572 . . 3  |-  ( ( C  e.  V  /\  ( F " C )  e.  _V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
131, 9, 11, 12syl3anc 1232 . 2  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  ~~  ( F " C ) )
1413ensymd 7606 1  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  ~~  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1844   _Vcvv 3061    C_ wss 3416   class class class wbr 4397   ran crn 4826    |` cres 4827   "cima 4828   -->wf 5567   -1-1->wf1 5568   -1-1-onto->wf1o 5570    ~~ cen 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-er 7350  df-en 7557
This theorem is referenced by:  ssenen  7731  phplem4  7739  fiint  7833  unxpwdom2  8050  znunithash  18903
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