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Theorem rdgeq1 8036
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Proof of Theorem rdgeq1
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6662 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(𝑔 dom 𝑔)) = (𝐺‘(𝑔 dom 𝑔)))
21ifeq2d 4482 . . . . 5 (𝐹 = 𝐺 → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))
32ifeq2d 4482 . . . 4 (𝐹 = 𝐺 → if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))
43mpteq2dv 5153 . . 3 (𝐹 = 𝐺 → (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
5 recseq 7999 . . 3 ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))) → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))))
64, 5syl 17 . 2 (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔)))))))
7 df-rdg 8035 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
8 df-rdg 8035 . 2 rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
96, 7, 83eqtr4g 2878 1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  Vcvv 3492  c0 4288  ifcif 4463   cuni 4830  cmpt 5137  dom cdm 5548  ran crn 5549  Lim wlim 6185  cfv 6348  recscrecs 7996  reccrdg 8034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-wrecs 7936  df-recs 7997  df-rdg 8035
This theorem is referenced by:  rdgeq12  8038  rdgsucmpt2  8055  frsucmpt2w  8064  frsucmpt2  8065  seqomlem0  8074  omv  8126  oev  8128  dffi3  8883  hsmex  9842  axdc  9931  seqeq2  13361  seqval  13368  trpredlem1  32963  trpredtr  32966  trpredmintr  32967  neibastop2  33606  rdgssun  34541  exrecfnlem  34542  dffinxpf  34548  finxpeq1  34549
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