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Theorem nfof 5987
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 5982 . 2  |-  oF R  =  ( u  e.  _V ,  v  e.  _V  |->  ( w  e.  ( dom  u  i^i  dom  v )  |->  ( ( u `  w
) R ( v `
 w ) ) ) )
2 nfcv 2281 . . 3  |-  F/_ x _V
3 nfcv 2281 . . . 4  |-  F/_ x
( dom  u  i^i  dom  v )
4 nfcv 2281 . . . . 5  |-  F/_ x
( u `  w
)
5 nfof.1 . . . . 5  |-  F/_ x R
6 nfcv 2281 . . . . 5  |-  F/_ x
( v `  w
)
74, 5, 6nfov 5801 . . . 4  |-  F/_ x
( ( u `  w ) R ( v `  w ) )
83, 7nfmpt 4020 . . 3  |-  F/_ x
( w  e.  ( dom  u  i^i  dom  v )  |->  ( ( u `  w ) R ( v `  w ) ) )
92, 2, 8nfmpo 5840 . 2  |-  F/_ x
( u  e.  _V ,  v  e.  _V  |->  ( w  e.  ( dom  u  i^i  dom  v
)  |->  ( ( u `
 w ) R ( v `  w
) ) ) )
101, 9nfcxfr 2278 1  |-  F/_ x  oF R
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2268   _Vcvv 2686    i^i cin 3070    |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774    e. cmpo 5776    oFcof 5980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982
This theorem is referenced by: (None)
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