Theorem nffrec 6223

Description: Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)

Hypotheses

Ref	Expression
nffrec.1	|- F/_ x F
nffrec.2	|- F/_ x A

Assertion

Ref	Expression
nffrec	|- F/_ xfrec ( F , A )

Proof of Theorem nffrec

Dummy variables g m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frec 6218	. 2 |- frec ( F , A ) = (recs ( ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } ) ) |` om )
2		nfcv 2240	. . . . 5 |- F/_ x _V
3		nfcv 2240	. . . . . . . 8 |- F/_ x om
4		nfv 1476	. . . . . . . . 9 |- F/ x dom g = suc m
5		nffrec.1	. . . . . . . . . . 11 |- F/_ x F
6		nfcv 2240	. . . . . . . . . . 11 |- F/_ x ( g ` m )
7	5, 6	nffv 5363	. . . . . . . . . 10 |- F/_ x ( F ` ( g ` m ) )
8	7	nfcri 2234	. . . . . . . . 9 |- F/ x y e. ( F ` ( g ` m ) )
9	4, 8	nfan 1512	. . . . . . . 8 |- F/ x ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
10	3, 9	nfrexya 2433	. . . . . . 7 |- F/ x E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
11		nfv 1476	. . . . . . . 8 |- F/ x dom g = (/)
12		nffrec.2	. . . . . . . . 9 |- F/_ x A
13	12	nfcri 2234	. . . . . . . 8 |- F/ x y e. A
14	11, 13	nfan 1512	. . . . . . 7 |- F/ x ( dom g = (/) /\ y e. A )
15	10, 14	nfor 1521	. . . . . 6 |- F/ x ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) )
16	15	nfab 2245	. . . . 5 |- F/_ x { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) }
17	2, 16	nfmpt 3960	. . . 4 |- F/_ x ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
18	17	nfrecs 6134	. . 3 |- F/_ xrecs ( ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } ) )
19	18, 3	nfres 4757	. 2 |- F/_ x (recs ( ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } ) ) |` om )
20	1, 19	nfcxfr 2237	1 |- F/_ xfrec ( F , A )

Syntax hints:   /\ wa 103   \/ wo 670   = wceq 1299   e. wcel 1448   {cab 2086   F/_wnfc 2227   E.wrex 2376   _Vcvv 2641   (/)c0 3310   |-> cmpt 3929   suc csuc 4225   omcom 4442   dom cdm 4477   |` cres 4479   ` cfv 5059  recscrecs 6131  freccfrec 6217

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-un 3025  df-in 3027  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-xp 4483  df-res 4489  df-iota 5024  df-fv 5067  df-recs 6132  df-frec 6218

This theorem is referenced by:  nfseq  10069