Theorem nfrecs 6322

Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
nfrecs.f	|- F/_ x F

Assertion

Ref	Expression
nfrecs	|- F/_ xrecs ( F )

Proof of Theorem nfrecs

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-recs 6320	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
2		nfcv 2329	. . . . 5 |- F/_ x On
3		nfv 1538	. . . . . 6 |- F/ x a Fn b
4		nfcv 2329	. . . . . . 7 |- F/_ x b
5		nfrecs.f	. . . . . . . . 9 |- F/_ x F
6		nfcv 2329	. . . . . . . . 9 |- F/_ x ( a |` c )
7	5, 6	nffv 5537	. . . . . . . 8 |- F/_ x ( F ` ( a |` c ) )
8	7	nfeq2 2341	. . . . . . 7 |- F/ x ( a ` c ) = ( F ` ( a |` c ) )
9	4, 8	nfralxy 2525	. . . . . 6 |- F/ x A. c e. b ( a ` c ) = ( F ` ( a |` c ) )
10	3, 9	nfan 1575	. . . . 5 |- F/ x ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
11	2, 10	nfrexxy 2526	. . . 4 |- F/ x E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
12	11	nfab 2334	. . 3 |- F/_ x { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
13	12	nfuni 3827	. 2 |- F/_ x U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
14	1, 13	nfcxfr 2326	1 |- F/_ xrecs ( F )

Syntax hints:   /\ wa 104   = wceq 1363   {cab 2173   F/_wnfc 2316   A.wral 2465   E.wrex 2466   U.cuni 3821   Oncon0 4375   |` cres 4640   Fn wfn 5223   ` cfv 5228  recscrecs 6319

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-recs 6320

This theorem is referenced by:  nffrec  6411