Theorem nfof 5899

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 5894	. 2 |- oF R = ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
2		nfcv 2235	. . 3 |- F/_ x _V
3		nfcv 2235	. . . 4 |- F/_ x ( dom u i^i dom v )
4		nfcv 2235	. . . . 5 |- F/_ x ( u ` w )
5		nfof.1	. . . . 5 |- F/_ x R
6		nfcv 2235	. . . . 5 |- F/_ x ( v ` w )
7	4, 5, 6	nfov 5717	. . . 4 |- F/_ x ( ( u ` w ) R ( v ` w ) )
8	3, 7	nfmpt 3952	. . 3 |- F/_ x ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) )
9	2, 2, 8	nfmpt2 5755	. 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 2232	1 |- F/_ x oF R

Syntax hints:   F/_wnfc 2222   _Vcvv 2633   i^i cin 3012   |-> cmpt 3921   dom cdm 4467   ` cfv 5049  (class class class)co 5690   |-> cmpt2 5692   oFcof 5892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-iota 5014  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894

This theorem is referenced by: (None)