Theorem nfrdg 7510

Description: Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Hypotheses

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

Assertion

Ref	Expression
nfrdg	|- F/_ x rec ( F , A )

Proof of Theorem nfrdg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rdg 7506	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
2		nfcv 2764	. . . 4 |- F/_ x _V
3		nfv 1843	. . . . 5 |- F/ x g = (/)
4		nfrdg.2	. . . . 5 |- F/_ x A
5		nfv 1843	. . . . . 6 |- F/ x Lim dom g
6		nfcv 2764	. . . . . 6 |- F/_ x U. ran g
7		nfrdg.1	. . . . . . 7 |- F/_ x F
8		nfcv 2764	. . . . . . 7 |- F/_ x ( g ` U. dom g )
9	7, 8	nffv 6198	. . . . . 6 |- F/_ x ( F ` ( g ` U. dom g ) )
10	5, 6, 9	nfif 4115	. . . . 5 |- F/_ x if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) )
11	3, 4, 10	nfif 4115	. . . 4 |- F/_ x if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) )
12	2, 11	nfmpt 4746	. . 3 |- F/_ x ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) )
13	12	nfrecs 7471	. 2 |- F/_ xrecs ( ( g e. _V |-> if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
14	1, 13	nfcxfr 2762	1 |- F/_ x rec ( F , A )

Syntax hints:   = wceq 1483   F/_wnfc 2751   _Vcvv 3200   (/)c0 3915   ifcif 4086   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Lim wlim 5724   ` cfv 5888  recscrecs 7467   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  rdgsucmptf  7524  rdgsucmptnf  7525  frsucmpt  7533  frsucmptn  7534  nfseq  12811  trpredlem1  31727  trpredrec  31738  finxpreclem6  33233