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Theorem relmpt2opab 6429
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
relmpt2opab.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )
Assertion
Ref Expression
relmpt2opab  |-  Rel  ( C F D )
Distinct variable groups:    x, w, y, z    y, B    x, A, y
Allowed substitution hints:    ph( x, y, z, w)    A( z, w)    B( x, z, w)    C( x, y, z, w)    D( x, y, z, w)    F( x, y, z, w)

Proof of Theorem relmpt2opab
StepHypRef Expression
1 relopab 5001 . . . . 5  |-  Rel  { <. z ,  w >.  | 
ph }
2 df-rel 4885 . . . . 5  |-  ( Rel 
{ <. z ,  w >.  |  ph }  <->  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 200 . . . 4  |-  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V )
43rgen2w 2774 . . 3  |-  A. x  e.  A  A. y  e.  B  { <. z ,  w >.  |  ph }  C_  ( _V  X.  _V )
5 relmpt2opab.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  { <. z ,  w >.  |  ph } )
65ovmptss 6428 . . 3  |-  ( A. x  e.  A  A. y  e.  B  { <. z ,  w >.  | 
ph }  C_  ( _V  X.  _V )  -> 
( C F D )  C_  ( _V  X.  _V ) )
74, 6ax-mp 8 . 2  |-  ( C F D )  C_  ( _V  X.  _V )
8 df-rel 4885 . 2  |-  ( Rel  ( C F D )  <->  ( C F D )  C_  ( _V  X.  _V ) )
97, 8mpbir 201 1  |-  Rel  ( C F D )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   A.wral 2705   _Vcvv 2956    C_ wss 3320   {copab 4265    X. cxp 4876   Rel wrel 4883  (class class class)co 6081    e. cmpt2 6083
This theorem is referenced by:  brovmpt2ex  6475  relfunc  14059  releqg  14987  releupa  21686  relwlk  28296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350
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