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Theorem nfof 6088
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  oF R

Proof of Theorem nfof
Dummy variables  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 6083 . 2  |-  oF R  =  ( u  e.  _V ,  v  e.  _V  |->  ( w  e.  ( dom  u  i^i  dom  v )  |->  ( ( u `  w
) R ( v `
 w ) ) ) )
2 nfcv 2319 . . 3  |-  F/_ x _V
3 nfcv 2319 . . . 4  |-  F/_ x
( dom  u  i^i  dom  v )
4 nfcv 2319 . . . . 5  |-  F/_ x
( u `  w
)
5 nfof.1 . . . . 5  |-  F/_ x R
6 nfcv 2319 . . . . 5  |-  F/_ x
( v `  w
)
74, 5, 6nfov 5905 . . . 4  |-  F/_ x
( ( u `  w ) R ( v `  w ) )
83, 7nfmpt 4096 . . 3  |-  F/_ x
( w  e.  ( dom  u  i^i  dom  v )  |->  ( ( u `  w ) R ( v `  w ) ) )
92, 2, 8nfmpo 5944 . 2  |-  F/_ x
( u  e.  _V ,  v  e.  _V  |->  ( w  e.  ( dom  u  i^i  dom  v
)  |->  ( ( u `
 w ) R ( v `  w
) ) ) )
101, 9nfcxfr 2316 1  |-  F/_ x  oF R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2306   _Vcvv 2738    i^i cin 3129    |-> cmpt 4065   dom cdm 4627   ` cfv 5217  (class class class)co 5875    e. cmpo 5877    oFcof 6081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-iota 5179  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083
This theorem is referenced by: (None)
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