MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrdg Structured version   Visualization version   GIF version

Theorem nfrdg 7374
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 𝑥𝐹
nfrdg.2 𝑥𝐴
Assertion
Ref Expression
nfrdg 𝑥rec(𝐹, 𝐴)

Proof of Theorem nfrdg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-rdg 7370 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2 nfcv 2750 . . . 4 𝑥V
3 nfv 1829 . . . . 5 𝑥 𝑔 = ∅
4 nfrdg.2 . . . . 5 𝑥𝐴
5 nfv 1829 . . . . . 6 𝑥Lim dom 𝑔
6 nfcv 2750 . . . . . 6 𝑥 ran 𝑔
7 nfrdg.1 . . . . . . 7 𝑥𝐹
8 nfcv 2750 . . . . . . 7 𝑥(𝑔 dom 𝑔)
97, 8nffv 6095 . . . . . 6 𝑥(𝐹‘(𝑔 dom 𝑔))
105, 6, 9nfif 4064 . . . . 5 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
113, 4, 10nfif 4064 . . . 4 𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
122, 11nfmpt 4668 . . 3 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1312nfrecs 7335 . 2 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
141, 13nfcxfr 2748 1 𝑥rec(𝐹, 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wnfc 2737  Vcvv 3172  c0 3873  ifcif 4035   cuni 4366  cmpt 4637  dom cdm 5028  ran crn 5029  Lim wlim 5627  cfv 5790  recscrecs 7331  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-iota 5754  df-fv 5798  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  rdgsucmptf  7388  rdgsucmptnf  7389  frsucmpt  7397  frsucmptn  7398  nfseq  12628  trpredlem1  30777  trpredrec  30788  finxpreclem6  32212
  Copyright terms: Public domain W3C validator