Theorem nffrec 6505

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 6500	. 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 2350	. . . . 5 |- F/_ x _V
3		nfcv 2350	. . . . . . . 8 |- F/_ x om
4		nfv 1552	. . . . . . . . 9 |- F/ x dom g = suc m
5		nffrec.1	. . . . . . . . . . 11 |- F/_ x F
6		nfcv 2350	. . . . . . . . . . 11 |- F/_ x ( g ` m )
7	5, 6	nffv 5609	. . . . . . . . . 10 |- F/_ x ( F ` ( g ` m ) )
8	7	nfcri 2344	. . . . . . . . 9 |- F/ x y e. ( F ` ( g ` m ) )
9	4, 8	nfan 1589	. . . . . . . 8 |- F/ x ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
10	3, 9	nfrexya 2549	. . . . . . 7 |- F/ x E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) )
11		nfv 1552	. . . . . . . 8 |- F/ x dom g = (/)
12		nffrec.2	. . . . . . . . 9 |- F/_ x A
13	12	nfcri 2344	. . . . . . . 8 |- F/ x y e. A
14	11, 13	nfan 1589	. . . . . . 7 |- F/ x ( dom g = (/) /\ y e. A )
15	10, 14	nfor 1598	. . . . . 6 |- F/ x ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) )
16	15	nfab 2355	. . . . 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 4152	. . . 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 6416	. . 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 4980	. 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 2347	1 |- F/_ xfrec ( F , A )

Syntax hints:   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2178   {cab 2193   F/_wnfc 2337   E.wrex 2487   _Vcvv 2776   (/)c0 3468   |-> cmpt 4121   suc csuc 4430   omcom 4656   dom cdm 4693   |` cres 4695   ` cfv 5290  recscrecs 6413  freccfrec 6499

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 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-in 3180  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-xp 4699  df-res 4705  df-iota 5251  df-fv 5298  df-recs 6414  df-frec 6500

This theorem is referenced by:  nfseq  10639