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Theorem nfof 6083
Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypothesis
Ref Expression
nfof.1  |-  F/_ x R
Assertion
Ref Expression
nfof  |-  F/_ x  o F R
Distinct variable group:    x, R

Proof of Theorem nfof
Dummy variables  f 
g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 6078 . 2  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( y  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  y
) R ( g `
 y ) ) ) )
2 nfcv 2419 . . 3  |-  F/_ x _V
3 nfcv 2419 . . . 4  |-  F/_ x
( dom  f  i^i  dom  g )
4 nfcv 2419 . . . . 5  |-  F/_ x
( f `  y
)
5 nfof.1 . . . . 5  |-  F/_ x R
6 nfcv 2419 . . . . 5  |-  F/_ x
( g `  y
)
74, 5, 6nfov 5881 . . . 4  |-  F/_ x
( ( f `  y ) R ( g `  y ) )
83, 7nfmpt 4108 . . 3  |-  F/_ x
( y  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  y ) R ( g `  y ) ) )
92, 2, 8nfmpt2 5916 . 2  |-  F/_ x
( f  e.  _V ,  g  e.  _V  |->  ( y  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  y ) R ( g `  y ) ) ) )
101, 9nfcxfr 2416 1  |-  F/_ x  o F R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2406   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076
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-rex 2549  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-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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