Theorem nfof 6298

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 6292	. 2 |- oF R = ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
2		nfcv 2392	. . 3 |- F/_ x _V
3		nfcv 2392	. . . 4 |- F/_ x ( dom u i^i dom v )
4		nfcv 2392	. . . . 5 |- F/_ x ( u ` w )
5		nfof.1	. . . . 5 |- F/_ x R
6		nfcv 2392	. . . . 5 |- F/_ x ( v ` w )
7	4, 5, 6	nfov 6105	. . . 4 |- F/_ x ( ( u ` w ) R ( v ` w ) )
8	3, 7	nfmpt 4218	. . 3 |- F/_ x ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) )
9	2, 2, 8	nfmpo 6147	. 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 2389	1 |- F/_ x oF R

Syntax hints:   F/_wnfc 2379   _Vcvv 2821   i^i cin 3219   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   oFcof 6290

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-iota 5332  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292

This theorem is referenced by: (None)