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Theorem nfofr 6084
Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypothesis
Ref Expression
nfof.1  |-  F/_ x R
Assertion
Ref Expression
nfofr  |-  F/_ x  o R R
Distinct variable group:    x, R

Proof of Theorem nfofr
Dummy variables  f 
g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofr 6079 . 2  |-  o R R  =  { <. f ,  g >.  |  A. y  e.  ( dom  f  i^i  dom  g )
( f `  y
) R ( g `
 y ) }
2 nfcv 2419 . . . 4  |-  F/_ x
( dom  f  i^i  dom  g )
3 nfcv 2419 . . . . 5  |-  F/_ x
( f `  y
)
4 nfof.1 . . . . 5  |-  F/_ x R
5 nfcv 2419 . . . . 5  |-  F/_ x
( g `  y
)
63, 4, 5nfbr 4067 . . . 4  |-  F/ x
( f `  y
) R ( g `
 y )
72, 6nfral 2596 . . 3  |-  F/ x A. y  e.  ( dom  f  i^i  dom  g
) ( f `  y ) R ( g `  y )
87nfopab 4084 . 2  |-  F/_ x { <. f ,  g
>.  |  A. y  e.  ( dom  f  i^i 
dom  g ) ( f `  y ) R ( g `  y ) }
91, 8nfcxfr 2416 1  |-  F/_ x  o R R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2406   A.wral 2543    i^i cin 3151   class class class wbr 4023   {copab 4076   dom cdm 4689   ` cfv 5255    o Rcofr 6077
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-ofr 6079
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