Theorem nfof 6164

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.

Step	Hyp	Ref	Expression
1		df-of 6158	. 2 |- oF R = ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
2		nfcv 2348	. . 3 |- F/_ x _V
3		nfcv 2348	. . . 4 |- F/_ x ( dom u i^i dom v )
4		nfcv 2348	. . . . 5 |- F/_ x ( u ` w )
5		nfof.1	. . . . 5 |- F/_ x R
6		nfcv 2348	. . . . 5 |- F/_ x ( v ` w )
7	4, 5, 6	nfov 5974	. . . 4 |- F/_ x ( ( u ` w ) R ( v ` w ) )
8	3, 7	nfmpt 4136	. . 3 |- F/_ x ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) )
9	2, 2, 8	nfmpo 6014	. 2 |- F/_ x ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
10	1, 9	nfcxfr 2345	1 |- F/_ x oF R

Syntax hints:   F/_wnfc 2335   _Vcvv 2772   i^i cin 3165   |-> cmpt 4105   dom cdm 4675   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   oFcof 6156

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-iota 5232  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-of 6158

This theorem is referenced by: (None)