Theorem nfrecs 6451

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 6449	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
2		nfcv 2372	. . . . 5 |- F/_ x On
3		nfv 1574	. . . . . 6 |- F/ x a Fn b
4		nfcv 2372	. . . . . . 7 |- F/_ x b
5		nfrecs.f	. . . . . . . . 9 |- F/_ x F
6		nfcv 2372	. . . . . . . . 9 |- F/_ x ( a |` c )
7	5, 6	nffv 5636	. . . . . . . 8 |- F/_ x ( F ` ( a |` c ) )
8	7	nfeq2 2384	. . . . . . 7 |- F/ x ( a ` c ) = ( F ` ( a |` c ) )
9	4, 8	nfralxy 2568	. . . . . 6 |- F/ x A. c e. b ( a ` c ) = ( F ` ( a |` c ) )
10	3, 9	nfan 1611	. . . . 5 |- F/ x ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
11	2, 10	nfrexw 2569	. . . 4 |- F/ x E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) )
12	11	nfab 2377	. . 3 |- F/_ x { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
13	12	nfuni 3893	. 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 2369	1 |- F/_ xrecs ( F )

Syntax hints:   /\ wa 104   = wceq 1395   {cab 2215   F/_wnfc 2359   A.wral 2508   E.wrex 2509   U.cuni 3887   Oncon0 4453   |` cres 4720   Fn wfn 5312   ` cfv 5317  recscrecs 6448

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-recs 6449

This theorem is referenced by:  nffrec  6540