Theorem nfrdg 8385

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 8381	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2		nfcv 2892	. . . 4 ⊢ Ⅎ𝑥V
3		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅
4		nfrdg.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔
6		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔
7		nfrdg.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔)
9	7, 8	nffv 6871	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔))
10	5, 6, 9	nfif 4522	. . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))
11	3, 4, 10	nfif 4522	. . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
12	2, 11	nfmpt 5208	. . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
13	12	nfrecs 8346	. 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
14	1, 13	nfcxfr 2890	1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Syntax hints:   = wceq 1540  Ⅎwnfc 2877  Vcvv 3450  ∅c0 4299  ifcif 4491  ∪ cuni 4874   ↦ cmpt 5191  dom cdm 5641  ran crn 5642  Lim wlim 6336  ‘cfv 6514  recscrecs 8342  reccrdg 8380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-iota 6467  df-fv 6522  df-ov 7393  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381

This theorem is referenced by:  rdgsucmptf  8399  rdgsucmptnf  8400  frsucmpt  8409  frsucmptn  8410  ttrclselem1  9685  ttrclselem2  9686  nfseq  13983  nfseqs  28188  rdgssun  37373  exrecfnlem  37374  finxpreclem6  37391