Theorem nfrdg 8345

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 8341	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥V
3		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅
4		nfrdg.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔
6		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔
7		nfrdg.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔)
9	7, 8	nffv 6844	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔))
10	5, 6, 9	nfif 4510	. . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))
11	3, 4, 10	nfif 4510	. . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
12	2, 11	nfmpt 5196	. . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
13	12	nfrecs 8306	. 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
14	1, 13	nfcxfr 2896	1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Syntax hints:   = wceq 1541  Ⅎwnfc 2883  Vcvv 3440  ∅c0 4285  ifcif 4479  ∪ cuni 4863   ↦ cmpt 5179  dom cdm 5624  ran crn 5625  Lim wlim 6318  ‘cfv 6492  recscrecs 8302  reccrdg 8340

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fv 6500  df-ov 7361  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  rdgsucmptf  8359  rdgsucmptnf  8360  frsucmpt  8369  frsucmptn  8370  ttrclselem1  9634  ttrclselem2  9635  nfseq  13934  nfseqs  28283  rdgssun  37579  exrecfnlem  37580  finxpreclem6  37597