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Theorem funimassov 6023
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 5566 . 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 5515 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5877 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3eqtr4di 2228 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eleq1d 2246 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4770 . 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, 6bitrdi 196 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455    C_ wss 3129   <.cop 3595    X. cxp 4624   dom cdm 4626   "cima 4629   Fun wfun 5210   ` cfv 5216  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877
This theorem is referenced by: (None)
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