Theorem nffrec 6451

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 6446	. 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 2336	. . . . 5 |- F/_ x _V
3		nfcv 2336	. . . . . . . 8 |- F/_ x om
4		nfv 1539	. . . . . . . . 9 |- F/ x dom g = suc m
5		nffrec.1	. . . . . . . . . . 11 |- F/_ x F
6		nfcv 2336	. . . . . . . . . . 11 |- F/_ x ( g ` m )
7	5, 6	nffv 5565	. . . . . . . . . 10 |- F/_ x ( F ` ( g ` m ) )
8	7	nfcri 2330	. . . . . . . . 9 |- F/ x y e. ( F ` ( g ` m ) )
9	4, 8	nfan 1576	. . . . . . . 8 |- F/ x ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
10	3, 9	nfrexya 2535	. . . . . . 7 |- F/ x E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
11		nfv 1539	. . . . . . . 8 |- F/ x dom g = (/)
12		nffrec.2	. . . . . . . . 9 |- F/_ x A
13	12	nfcri 2330	. . . . . . . 8 |- F/ x y e. A
14	11, 13	nfan 1576	. . . . . . 7 |- F/ x ( dom g = (/) /\ y e. A )
15	10, 14	nfor 1585	. . . . . 6 |- F/ x ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) )
16	15	nfab 2341	. . . . 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 4122	. . . 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 6362	. . 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 4945	. 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 2333	1 |- F/_ xfrec ( F , A )

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   {cab 2179   F/_wnfc 2323   E.wrex 2473   _Vcvv 2760   (/)c0 3447   |-> cmpt 4091   suc csuc 4397   omcom 4623   dom cdm 4660   |` cres 4662   ` cfv 5255  recscrecs 6359  freccfrec 6445

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-in 3160  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-xp 4666  df-res 4672  df-iota 5216  df-fv 5263  df-recs 6360  df-frec 6446

This theorem is referenced by:  nfseq  10531