MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1imaeng Unicode version

Theorem f1imaeng 6921
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
f1imaeng  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )

Proof of Theorem f1imaeng
StepHypRef Expression
1 f1ores 5487 . . . 4  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
2 f1oeng 6880 . . . . 5  |-  ( ( C  e.  V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
32ancoms 439 . . . 4  |-  ( ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  /\  C  e.  V )  ->  C  ~~  ( F " C
) )
41, 3sylan 457 . . 3  |-  ( ( ( F : A -1-1-> B  /\  C  C_  A
)  /\  C  e.  V )  ->  C  ~~  ( F " C
) )
543impa 1146 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  C  ~~  ( F
" C ) )
6 ensym 6910 . 2  |-  ( C 
~~  ( F " C )  ->  ( F " C )  ~~  C )
75, 6syl 15 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   class class class wbr 4023    |` cres 4691   "cima 4692   -1-1->wf1 5252   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  f1imaen  6923  ackbij1b  7865  enfin1ai  8010  isercolllem2  12139  ballotlemro  23081  pmtrfconj  27407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864
  Copyright terms: Public domain W3C validator