Theorem nfrdg 8402

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 8398	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2		nfcv 2925	. . . 4 ⊢ Ⅎ𝑥V
3		nfv 1944	. . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅
4		nfrdg.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
5		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔
6		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔
7		nfrdg.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
8		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔)
9	7, 8	nffv 6893	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔))
10	5, 6, 9	nfif 4519	. . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))
11	3, 4, 10	nfif 4519	. . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))
12	2, 11	nfmpt 5210	. . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
13	12	nfrecs 8362	. 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
14	1, 13	nfcxfr 2923	1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴)

Syntax hints:   = wceq 1570  Ⅎwnfc 2910  Vcvv 3455  ∅c0 4287  ifcif 4488  ∪ cuni 4873   ↦ cmpt 5193  dom cdm 5663  ran crn 5664  Lim wlim 6363  ‘cfv 6538  recscrecs 8358  reccrdg 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7415  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398

This theorem is referenced by:  rdgsucmptf  8416  rdgsucmptnf  8417  frsucmpt  8426  frsucmptn  8427  ttrclselem1  9695  ttrclselem2  9696  nfseq  14049  nfseqs  28461  rdgssun  38005  exrecfnlem  38006  finxpreclem6  38023