Theorem nfofr 6142

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 6136	. 2 |- oR R = { <. u , v >. | A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w ) }
2		nfcv 2339	. . . 4 |- F/_ x ( dom u i^i dom v )
3		nfcv 2339	. . . . 5 |- F/_ x ( u ` w )
4		nfof.1	. . . . 5 |- F/_ x R
5		nfcv 2339	. . . . 5 |- F/_ x ( v ` w )
6	3, 4, 5	nfbr 4079	. . . 4 |- F/ x ( u ` w ) R ( v ` w )
7	2, 6	nfralxy 2535	. . 3 |- F/ x A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w )
8	7	nfopab 4101	. 2 |- F/_ x { <. u , v >. | A. w e. ( dom u i^i dom v ) ( u ` w ) R ( v ` w ) }
9	1, 8	nfcxfr 2336	1 |- F/_ x oR R

Syntax hints:   F/_wnfc 2326   A.wral 2475   i^i cin 3156   class class class wbr 4033   {copab 4093   dom cdm 4663   ` cfv 5258   oRcofr 6134

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-ofr 6136

This theorem is referenced by: (None)