Theorem nfof 6236

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 6230	. 2 |- oF R = ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
2		nfcv 2372	. . 3 |- F/_ x _V
3		nfcv 2372	. . . 4 |- F/_ x ( dom u i^i dom v )
4		nfcv 2372	. . . . 5 |- F/_ x ( u ` w )
5		nfof.1	. . . . 5 |- F/_ x R
6		nfcv 2372	. . . . 5 |- F/_ x ( v ` w )
7	4, 5, 6	nfov 6043	. . . 4 |- F/_ x ( ( u ` w ) R ( v ` w ) )
8	3, 7	nfmpt 4179	. . 3 |- F/_ x ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) )
9	2, 2, 8	nfmpo 6085	. 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 2369	1 |- F/_ x oF R

Syntax hints:   F/_wnfc 2359   _Vcvv 2800   i^i cin 3197   |-> cmpt 4148   dom cdm 4723   ` cfv 5324  (class class class)co 6013   e. cmpo 6015   oFcof 6228

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-iota 5284  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by: (None)