ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1imaen2g Unicode version

Theorem f1imaen2g 6793
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6794 does not need ax-setind 4537.) (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 531 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  e.  V )
2 simplr 528 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  B  e.  V )
3 f1f 5422 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
4 imassrn 4982 . . . . . . 7  |-  ( F
" C )  C_  ran  F
5 frn 5375 . . . . . . 7  |-  ( F : A --> B  ->  ran  F  C_  B )
64, 5sstrid 3167 . . . . . 6  |-  ( F : A --> B  -> 
( F " C
)  C_  B )
73, 6syl 14 . . . . 5  |-  ( F : A -1-1-> B  -> 
( F " C
)  C_  B )
87ad2antrr 488 . . . 4  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  C_  B )
92, 8ssexd 4144 . . 3  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  -> 
( F " C
)  e.  _V )
10 f1ores 5477 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
1110ad2ant2r 509 . . 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 6755 . . 3  |-  ( ( C  e.  V  /\  ( F " C )  e.  _V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
131, 9, 11, 12syl3anc 1238 . 2  |-  ( ( ( F : A -1-1-> B  /\  B  e.  V
)  /\  ( C  C_  A  /\  C  e.  V ) )  ->  C  ~~  ( F " C ) )
1413ensymd 6783 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 104    e. wcel 2148   _Vcvv 2738    C_ wss 3130   class class class wbr 4004   ran crn 4628    |` cres 4629   "cima 4630   -->wf 5213   -1-1->wf1 5214   -1-1-onto->wf1o 5216    ~~ cen 6738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-er 6535  df-en 6741
This theorem is referenced by:  ssenen  6851  phplem4  6855  phplem4dom  6862  phplem4on  6867  fiintim  6928
  Copyright terms: Public domain W3C validator