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Theorem ovelimab 5998
Description: Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
Assertion
Ref Expression
ovelimab  |-  ( ( F  Fn  A  /\  ( B  X.  C
)  C_  A )  ->  ( D  e.  ( F " ( B  X.  C ) )  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y   
x, F, y

Proof of Theorem ovelimab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fvelimab 5578 . 2  |-  ( ( F  Fn  A  /\  ( B  X.  C
)  C_  A )  ->  ( D  e.  ( F " ( B  X.  C ) )  <->  E. z  e.  ( B  X.  C ) ( F `  z )  =  D ) )
2 fveq2 5525 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5861 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2333 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eqeq1d 2291 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  ( x F y )  =  D ) )
6 eqcom 2285 . . . 4  |-  ( ( x F y )  =  D  <->  D  =  ( x F y ) )
75, 6syl6bb 252 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  D  =  (
x F y ) ) )
87rexxp 4828 . 2  |-  ( E. z  e.  ( B  X.  C ) ( F `  z )  =  D  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) )
91, 8syl6bb 252 1  |-  ( ( F  Fn  A  /\  ( B  X.  C
)  C_  A )  ->  ( D  e.  ( F " ( B  X.  C ) )  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   <.cop 3643    X. cxp 4687   "cima 4692    Fn wfn 5250   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  dfz2  10041  elq  10318  shsel  21893  ofrn2  23207
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  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-fv 5263  df-ov 5861
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