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Theorem f1imaen2g 7568
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7569 does not need ax-reg 8009.) (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 756 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  e.  V )
2 simplr 754 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  B  e.  V )
3 f1f 5774 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
4 imassrn 5341 . . . . . . 7  |-  ( F
" C )  C_  ran  F
5 frn 5730 . . . . . . 7  |-  ( F : A --> B  ->  ran  F  C_  B )
64, 5syl5ss 3510 . . . . . 6  |-  ( F : A --> B  -> 
( F " C
)  C_  B )
73, 6syl 16 . . . . 5  |-  ( F : A -1-1-> B  -> 
( F " C
)  C_  B )
87ad2antrr 725 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  C_  B )
92, 8ssexd 4589 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  e.  _V )
10 f1ores 5823 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
1110ad2ant2r 746 . . 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 7524 . . 3  |-  ( ( C  e.  V  /\  ( F " C )  e.  _V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
131, 9, 11, 12syl3anc 1223 . 2  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  ~~  ( F " C ) )
1413ensymd 7558 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 1762   _Vcvv 3108    C_ wss 3471   class class class wbr 4442   ran crn 4995    |` cres 4996   "cima 4997   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580    ~~ cen 7505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-er 7303  df-en 7509
This theorem is referenced by:  ssenen  7683  phplem4  7691  fiint  7788  unxpwdom2  8005  znunithash  18365
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