Theorem nfrecs 6411

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 6409	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
2		nfcv 2349	. . . . 5 |- F/_ x On
3		nfv 1552	. . . . . 6 |- F/ x a Fn b
4		nfcv 2349	. . . . . . 7 |- F/_ x b
5		nfrecs.f	. . . . . . . . 9 |- F/_ x F
6		nfcv 2349	. . . . . . . . 9 |- F/_ x ( a |` c )
7	5, 6	nffv 5604	. . . . . . . 8 |- F/_ x ( F ` ( a |` c ) )
8	7	nfeq2 2361	. . . . . . 7 |- F/ x ( a ` c ) = ( F ` ( a |` c ) )
9	4, 8	nfralxy 2545	. . . . . 6 |- F/ x A. c e. b ( a ` c ) = ( F ` ( a |` c ) )
10	3, 9	nfan 1589	. . . . 5 |- F/ x ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
11	2, 10	nfrexw 2546	. . . 4 |- F/ x E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
12	11	nfab 2354	. . 3 |- F/_ x { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
13	12	nfuni 3865	. 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 2346	1 |- F/_ xrecs ( F )

Syntax hints:   /\ wa 104   = wceq 1373   {cab 2192   F/_wnfc 2336   A.wral 2485   E.wrex 2486   U.cuni 3859   Oncon0 4423   |` cres 4690   Fn wfn 5280   ` cfv 5285  recscrecs 6408

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 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-iota 5246  df-fv 5293  df-recs 6409

This theorem is referenced by:  nffrec  6500