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Theorem funimassov 5997
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 5573 . 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 5525 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5861 . . . . 5  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2333 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( x F y ) )
54eleq1d 2349 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4827 . 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 252 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  dprd2da  15277  xkococnlem  17353  iscfil2  18692  itg1addlem4  19054  issh2  21788  cvmlift2lem9  23842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
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