Theorem nffrec 6561

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 6556	. 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 2374	. . . . 5 |- F/_ x _V
3		nfcv 2374	. . . . . . . 8 |- F/_ x om
4		nfv 1576	. . . . . . . . 9 |- F/ x dom g = suc m
5		nffrec.1	. . . . . . . . . . 11 |- F/_ x F
6		nfcv 2374	. . . . . . . . . . 11 |- F/_ x ( g ` m )
7	5, 6	nffv 5649	. . . . . . . . . 10 |- F/_ x ( F ` ( g ` m ) )
8	7	nfcri 2368	. . . . . . . . 9 |- F/ x y e. ( F ` ( g ` m ) )
9	4, 8	nfan 1613	. . . . . . . 8 |- F/ x ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
10	3, 9	nfrexya 2573	. . . . . . 7 |- F/ x E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
11		nfv 1576	. . . . . . . 8 |- F/ x dom g = (/)
12		nffrec.2	. . . . . . . . 9 |- F/_ x A
13	12	nfcri 2368	. . . . . . . 8 |- F/ x y e. A
14	11, 13	nfan 1613	. . . . . . 7 |- F/ x ( dom g = (/) /\ y e. A )
15	10, 14	nfor 1622	. . . . . 6 |- F/ x ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) )
16	15	nfab 2379	. . . . 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 4181	. . . 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 6472	. . 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 5015	. 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 2371	1 |- F/_ xfrec ( F , A )

Syntax hints:   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   F/_wnfc 2361   E.wrex 2511   _Vcvv 2802   (/)c0 3494   |-> cmpt 4150   suc csuc 4462   omcom 4688   dom cdm 4725   |` cres 4727   ` cfv 5326  recscrecs 6469  freccfrec 6555

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-res 4737  df-iota 5286  df-fv 5334  df-recs 6470  df-frec 6556

This theorem is referenced by:  nfseq  10718