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Theorem ovelimab 5679
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 5257 . 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 5206 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5543 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2106 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eqeq1d 2064 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  ( x F y )  =  D ) )
6 eqcom 2058 . . . 4  |-  ( ( x F y )  =  D  <->  D  =  ( x F y ) )
75, 6syl6bb 189 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  D  <->  D  =  (
x F y ) ) )
87rexxp 4508 . 2  |-  ( E. z  e.  ( B  X.  C ) ( F `  z )  =  D  <->  E. x  e.  B  E. y  e.  C  D  =  ( x F y ) )
91, 8syl6bb 189 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    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324    C_ wss 2945   <.cop 3406    X. cxp 4371   "cima 4376    Fn wfn 4925   ` cfv 4930  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938  df-ov 5543
This theorem is referenced by:  dfz2  8371  elq  8654
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