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Theorem nfofr 6100
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 6095 . 2  |-  o R R  =  { <. f ,  g >.  |  A. y  e.  ( dom  f  i^i  dom  g )
( f `  y
) R ( g `
 y ) }
2 nfcv 2432 . . . 4  |-  F/_ x
( dom  f  i^i  dom  g )
3 nfcv 2432 . . . . 5  |-  F/_ x
( f `  y
)
4 nfof.1 . . . . 5  |-  F/_ x R
5 nfcv 2432 . . . . 5  |-  F/_ x
( g `  y
)
63, 4, 5nfbr 4083 . . . 4  |-  F/ x
( f `  y
) R ( g `
 y )
72, 6nfral 2609 . . 3  |-  F/ x A. y  e.  ( dom  f  i^i  dom  g
) ( f `  y ) R ( g `  y )
87nfopab 4100 . 2  |-  F/_ x { <. f ,  g
>.  |  A. y  e.  ( dom  f  i^i 
dom  g ) ( f `  y ) R ( g `  y ) }
91, 8nfcxfr 2429 1  |-  F/_ x  o R R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2419   A.wral 2556    i^i cin 3164   class class class wbr 4039   {copab 4092   dom cdm 4705   ` cfv 5271    o Rcofr 6093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-ofr 6095
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