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Theorem nfof 6099
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 6094 . 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 2432 . . 3  |-  F/_ x _V
3 nfcv 2432 . . . 4  |-  F/_ x
( dom  f  i^i  dom  g )
4 nfcv 2432 . . . . 5  |-  F/_ x
( f `  y
)
5 nfof.1 . . . . 5  |-  F/_ x R
6 nfcv 2432 . . . . 5  |-  F/_ x
( g `  y
)
74, 5, 6nfov 5897 . . . 4  |-  F/_ x
( ( f `  y ) R ( g `  y ) )
83, 7nfmpt 4124 . . 3  |-  F/_ x
( y  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  y ) R ( g `  y ) ) )
92, 2, 8nfmpt2 5932 . 2  |-  F/_ x
( f  e.  _V ,  g  e.  _V  |->  ( y  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  y ) R ( g `  y ) ) ) )
101, 9nfcxfr 2429 1  |-  F/_ x  o F R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2419   _Vcvv 2801    i^i cin 3164    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    o Fcof 6092
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-rex 2562  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-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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