Theorem nfrdg 8052

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	⊢ Ⅎ𝑥𝐹
nfrdg.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfrdg	⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Proof of Theorem nfrdg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rdg 8048	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2		nfcv 2979	. . . 4 ⊢ Ⅎ𝑥V
3		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅
4		nfrdg.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔
6		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔
7		nfrdg.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔)
9	7, 8	nffv 6682	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔))
10	5, 6, 9	nfif 4498	. . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))
11	3, 4, 10	nfif 4498	. . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
12	2, 11	nfmpt 5165	. . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
13	12	nfrecs 8013	. 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
14	1, 13	nfcxfr 2977	1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Syntax hints:   = wceq 1537  Ⅎwnfc 2963  Vcvv 3496  ∅c0 4293  ifcif 4469  ∪ cuni 4840   ↦ cmpt 5148  dom cdm 5557  ran crn 5558  Lim wlim 6194  ‘cfv 6357  recscrecs 8009  reccrdg 8047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-iota 6316  df-fv 6365  df-wrecs 7949  df-recs 8010  df-rdg 8048

This theorem is referenced by:  rdgsucmptf  8066  rdgsucmptnf  8067  frsucmpt  8075  frsucmptn  8076  nfseq  13382  trpredlem1  33068  trpredrec  33079  rdgssun  34661  exrecfnlem  34662  finxpreclem6  34679