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Theorem ovelimab 6262
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 5768 . 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 5712 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6115 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2493 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eqeq1d 2451 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  ( x F y )  =  D ) )
6 eqcom 2445 . . . 4  |-  ( ( x F y )  =  D  <->  D  =  ( x F y ) )
75, 6syl6bb 261 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  D  =  (
x F y ) ) )
87rexxp 5003 . 2  |-  ( E. z  e.  ( B  X.  C ) ( F `  z )  =  D  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) )
91, 8syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737    C_ wss 3349   <.cop 3904    X. cxp 4859   "cima 4864    Fn wfn 5434   ` cfv 5439  (class class class)co 6112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447  df-ov 6115
This theorem is referenced by:  dfz2  10685  elq  10976  shsel  24739  ofrn2  25980  eulerpartlemgh  26783
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