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Theorem relmptopab 6294
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 5815 . . 3  |-  ( A. x  e.  A  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )  -> 
( F `  B
)  C_  ( _V  X.  _V ) )
3 relopab 5003 . . . . 5  |-  Rel  { <. y ,  z >.  |  ph }
4 df-rel 4887 . . . . 5  |-  ( Rel 
{ <. y ,  z
>.  |  ph }  <->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
53, 4mpbi 201 . . . 4  |-  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )
65a1i 11 . . 3  |-  ( x  e.  A  ->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
72, 6mprg 2777 . 2  |-  ( F `
 B )  C_  ( _V  X.  _V )
8 df-rel 4887 . 2  |-  ( Rel  ( F `  B
)  <->  ( F `  B )  C_  ( _V  X.  _V ) )
97, 8mpbir 202 1  |-  Rel  ( F `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {copab 4267    e. cmpt 4268    X. cxp 4878   Rel wrel 4885   ` cfv 5456
This theorem is referenced by:  reldvdsr  15751  lmrel  17296  phtpcrel  19020  ulmrel  20296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464
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