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Theorem funimassov 5650
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
funimassov  |-  ( ( Fun  F  /\  ( A  X.  B )  C_  dom  F )  ->  (
( F " ( A  X.  B ) ) 
C_  C  <->  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, F, y

Proof of Theorem funimassov
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 funimass4 5224 . 2  |-  ( ( Fun  F  /\  ( A  X.  B )  C_  dom  F )  ->  (
( F " ( A  X.  B ) ) 
C_  C  <->  A. z  e.  ( A  X.  B
) ( F `  z )  e.  C
) )
2 fveq2 5178 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5515 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2090 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eleq1d 2106 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4479 . 2  |-  ( A. z  e.  ( A  X.  B ) ( F `
 z )  e.  C  <->  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )
71, 6syl6bb 185 1  |-  ( ( Fun  F  /\  ( A  X.  B )  C_  dom  F )  ->  (
( F " ( A  X.  B ) ) 
C_  C  <->  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306    C_ wss 2917   <.cop 3378    X. cxp 4343   dom cdm 4345   "cima 4348   Fun wfun 4896   ` cfv 4902  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515
This theorem is referenced by: (None)
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