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Theorem ovelimab 6428
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 5914 . 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 5857 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6278 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2519 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eqeq1d 2462 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  ( x F y )  =  D ) )
6 eqcom 2469 . . . 4  |-  ( ( x F y )  =  D  <->  D  =  ( x F y ) )
75, 6syl6bb 261 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  D  =  (
x F y ) ) )
87rexxp 5136 . 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 1374    e. wcel 1762   E.wrex 2808    C_ wss 3469   <.cop 4026    X. cxp 4990   "cima 4995    Fn wfn 5574   ` cfv 5579  (class class class)co 6275
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-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-ov 6278
This theorem is referenced by:  dfz2  10871  elq  11173  shsel  25894  ofrn2  27139  eulerpartlemgh  27943
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