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Theorem relmptopab 6065
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
relmptopab.1  |-  F  =  ( x  e.  A  |->  { <. y ,  z
>.  |  ph } )
Assertion
Ref Expression
relmptopab  |-  Rel  ( F `  B )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x, y, z)    A( y, z)    B( x, y, z)    F( x, y, z)

Proof of Theorem relmptopab
StepHypRef Expression
1 relmptopab.1 . . . 4  |-  F  =  ( x  e.  A  |->  { <. y ,  z
>.  |  ph } )
21fvmptss 5609 . . 3  |-  ( A. x  e.  A  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )  -> 
( F `  B
)  C_  ( _V  X.  _V ) )
3 relopab 4812 . . . . 5  |-  Rel  { <. y ,  z >.  |  ph }
4 df-rel 4696 . . . . 5  |-  ( Rel 
{ <. y ,  z
>.  |  ph }  <->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
53, 4mpbi 199 . . . 4  |-  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )
65a1i 10 . . 3  |-  ( x  e.  A  ->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
72, 6mprg 2612 . 2  |-  ( F `
 B )  C_  ( _V  X.  _V )
8 df-rel 4696 . 2  |-  ( Rel  ( F `  B
)  <->  ( F `  B )  C_  ( _V  X.  _V ) )
97, 8mpbir 200 1  |-  Rel  ( F `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {copab 4076    e. cmpt 4077    X. cxp 4687   Rel wrel 4694   ` cfv 5255
This theorem is referenced by:  reldvdsr  15426  lmrel  16960  phtpcrel  18491  ulmrel  19757
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-13 1686  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-pow 4188  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-br 4024  df-opab 4078  df-mpt 4079  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-fv 5263
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