Theorem nfof 6243

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 6237	. 2 |- oF R = ( u e. _V , v e. _V |-> ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) ) )
2		nfcv 2373	. . 3 |- F/_ x _V
3		nfcv 2373	. . . 4 |- F/_ x ( dom u i^i dom v )
4		nfcv 2373	. . . . 5 |- F/_ x ( u ` w )
5		nfof.1	. . . . 5 |- F/_ x R
6		nfcv 2373	. . . . 5 |- F/_ x ( v ` w )
7	4, 5, 6	nfov 6050	. . . 4 |- F/_ x ( ( u ` w ) R ( v ` w ) )
8	3, 7	nfmpt 4180	. . 3 |- F/_ x ( w e. ( dom u i^i dom v ) |-> ( ( u ` w ) R ( v ` w ) ) )
9	2, 2, 8	nfmpo 6092	. 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 2370	1 |- F/_ x oF R

Syntax hints:   F/_wnfc 2360   _Vcvv 2801   i^i cin 3198   |-> cmpt 4149   dom cdm 4724   ` cfv 5325  (class class class)co 6020   e. cmpo 6022   oFcof 6235

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-un 3203  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-br 4088  df-opab 4150  df-mpt 4151  df-iota 5285  df-fv 5333  df-ov 6023  df-oprab 6024  df-mpo 6025  df-of 6237

This theorem is referenced by: (None)