Theorem nfrecs 6516

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 6514	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
2		nfcv 2375	. . . . 5 |- F/_ x On
3		nfv 1577	. . . . . 6 |- F/ x a Fn b
4		nfcv 2375	. . . . . . 7 |- F/_ x b
5		nfrecs.f	. . . . . . . . 9 |- F/_ x F
6		nfcv 2375	. . . . . . . . 9 |- F/_ x ( a |` c )
7	5, 6	nffv 5658	. . . . . . . 8 |- F/_ x ( F ` ( a |` c ) )
8	7	nfeq2 2387	. . . . . . 7 |- F/ x ( a ` c ) = ( F ` ( a |` c ) )
9	4, 8	nfralxy 2571	. . . . . 6 |- F/ x A. c e. b ( a ` c ) = ( F ` ( a |` c ) )
10	3, 9	nfan 1614	. . . . 5 |- F/ x ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
11	2, 10	nfrexw 2572	. . . 4 |- F/ x E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
12	11	nfab 2380	. . 3 |- F/_ x { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
13	12	nfuni 3904	. 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 2372	1 |- F/_ xrecs ( F )

Syntax hints:   /\ wa 104   = wceq 1398   {cab 2217   F/_wnfc 2362   A.wral 2511   E.wrex 2512   U.cuni 3898   Oncon0 4466   |` cres 4733   Fn wfn 5328   ` cfv 5333  recscrecs 6513

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-recs 6514

This theorem is referenced by:  nffrec  6605