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Theorem nfrdg 7909
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 7905 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2 nfcv 2951 . . . 4 𝑥V
3 nfv 1896 . . . . 5 𝑥 𝑔 = ∅
4 nfrdg.2 . . . . 5 𝑥𝐴
5 nfv 1896 . . . . . 6 𝑥Lim dom 𝑔
6 nfcv 2951 . . . . . 6 𝑥 ran 𝑔
7 nfrdg.1 . . . . . . 7 𝑥𝐹
8 nfcv 2951 . . . . . . 7 𝑥(𝑔 dom 𝑔)
97, 8nffv 6555 . . . . . 6 𝑥(𝐹‘(𝑔 dom 𝑔))
105, 6, 9nfif 4416 . . . . 5 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
113, 4, 10nfif 4416 . . . 4 𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
122, 11nfmpt 5064 . . 3 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1312nfrecs 7870 . 2 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
141, 13nfcxfr 2949 1 𝑥rec(𝐹, 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1525  wnfc 2935  Vcvv 3440  c0 4217  ifcif 4387   cuni 4751  cmpt 5047  dom cdm 5450  ran crn 5451  Lim wlim 6074  cfv 6232  recscrecs 7866  reccrdg 7904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-xp 5456  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-iota 6196  df-fv 6240  df-wrecs 7805  df-recs 7867  df-rdg 7905
This theorem is referenced by:  rdgsucmptf  7923  rdgsucmptnf  7924  frsucmpt  7932  frsucmptn  7933  nfseq  13233  trpredlem1  32677  trpredrec  32688  rdgssun  34211  exrecfnlem  34212  finxpreclem6  34229
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