Theorem nfrecs 6392

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 6390	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
2		nfcv 2347	. . . . 5 |- F/_ x On
3		nfv 1550	. . . . . 6 |- F/ x a Fn b
4		nfcv 2347	. . . . . . 7 |- F/_ x b
5		nfrecs.f	. . . . . . . . 9 |- F/_ x F
6		nfcv 2347	. . . . . . . . 9 |- F/_ x ( a |` c )
7	5, 6	nffv 5585	. . . . . . . 8 |- F/_ x ( F ` ( a |` c ) )
8	7	nfeq2 2359	. . . . . . 7 |- F/ x ( a ` c ) = ( F ` ( a |` c ) )
9	4, 8	nfralxy 2543	. . . . . 6 |- F/ x A. c e. b ( a ` c ) = ( F ` ( a |` c ) )
10	3, 9	nfan 1587	. . . . 5 |- F/ x ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
11	2, 10	nfrexw 2544	. . . 4 |- F/ x E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
12	11	nfab 2352	. . 3 |- F/_ x { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
13	12	nfuni 3855	. 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 2344	1 |- F/_ xrecs ( F )

Syntax hints:   /\ wa 104   = wceq 1372   {cab 2190   F/_wnfc 2334   A.wral 2483   E.wrex 2484   U.cuni 3849   Oncon0 4409   |` cres 4676   Fn wfn 5265   ` cfv 5270  recscrecs 6389

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-recs 6390

This theorem is referenced by:  nffrec  6481