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Theorem relmptopab 6544
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 5981 . . 3  |-  ( A. x  e.  A  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )  -> 
( F `  B
)  C_  ( _V  X.  _V ) )
3 relopab 4979 . . . . 5  |-  Rel  { <. y ,  z >.  |  ph }
4 df-rel 4860 . . . . 5  |-  ( Rel 
{ <. y ,  z
>.  |  ph }  <->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
53, 4mpbi 213 . . . 4  |-  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V )
65a1i 11 . . 3  |-  ( x  e.  A  ->  { <. y ,  z >.  |  ph }  C_  ( _V  X.  _V ) )
72, 6mprg 2763 . 2  |-  ( F `
 B )  C_  ( _V  X.  _V )
8 df-rel 4860 . 2  |-  ( Rel  ( F `  B
)  <->  ( F `  B )  C_  ( _V  X.  _V ) )
97, 8mpbir 214 1  |-  Rel  ( F `  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    e. wcel 1898   _Vcvv 3057    C_ wss 3416   {copab 4474    |-> cmpt 4475    X. cxp 4851   Rel wrel 4858   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  reldvdsr  17921  lmrel  20295  phtpcrel  22073  ulmrel  23382  ercgrg  24611
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