Theorem nfofr 6188

Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypothesis

Ref	Expression
nfof.1	|- F/_ x R

Assertion

Ref	Expression
nfofr	|- F/_ x oR R

Proof of Theorem nfofr

Dummy variables u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofr 6182	. 2 |- oR R = { <. u , v >. | A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w ) }
2		nfcv 2350	. . . 4 |- F/_ x ( dom u i^i dom v )
3		nfcv 2350	. . . . 5 |- F/_ x ( u ` w )
4		nfof.1	. . . . 5 |- F/_ x R
5		nfcv 2350	. . . . 5 |- F/_ x ( v ` w )
6	3, 4, 5	nfbr 4106	. . . 4 |- F/ x ( u ` w ) R ( v ` w )
7	2, 6	nfralxy 2546	. . 3 |- F/ x A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w )
8	7	nfopab 4128	. 2 |- F/_ x { <. u , v >. | A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w ) }
9	1, 8	nfcxfr 2347	1 |- F/_ x oR R

Syntax hints:   F/_wnfc 2337   A.wral 2486   i^i cin 3173   class class class wbr 4059   {copab 4120   dom cdm 4693   ` cfv 5290   oRcofr 6180

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-ofr 6182

This theorem is referenced by: (None)