Theorem nfrdg 8470

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 8466	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥V
3		nfv 1913	. . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅
4		nfrdg.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔
6		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔
7		nfrdg.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔)
9	7, 8	nffv 6930	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔))
10	5, 6, 9	nfif 4578	. . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))
11	3, 4, 10	nfif 4578	. . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
12	2, 11	nfmpt 5273	. . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
13	12	nfrecs 8431	. 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
14	1, 13	nfcxfr 2906	1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Syntax hints:   = wceq 1537  Ⅎwnfc 2893  Vcvv 3488  ∅c0 4352  ifcif 4548  ∪ cuni 4931   ↦ cmpt 5249  dom cdm 5700  ran crn 5701  Lim wlim 6396  ‘cfv 6573  recscrecs 8426  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fv 6581  df-ov 7451  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  rdgsucmptf  8484  rdgsucmptnf  8485  frsucmpt  8494  frsucmptn  8495  ttrclselem1  9794  ttrclselem2  9795  nfseq  14062  nfseqs  28311  rdgssun  37344  exrecfnlem  37345  finxpreclem6  37362