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Theorem rdgeq1 7753
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 6417 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(𝑔 dom 𝑔)) = (𝐺‘(𝑔 dom 𝑔)))
21ifeq2d 4309 . . . . 5 (𝐹 = 𝐺 → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))
32ifeq2d 4309 . . . 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 4950 . . 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 7716 . . 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 7752 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
8 df-rdg 7752 . 2 rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
96, 7, 83eqtr4g 2876 1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  Vcvv 3402  c0 4127  ifcif 4290   cuni 4641  cmpt 4934  dom cdm 5324  ran crn 5325  Lim wlim 5951  cfv 6111  recscrecs 7713  reccrdg 7751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-xp 5330  df-cnv 5332  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-iota 6074  df-fv 6119  df-wrecs 7652  df-recs 7714  df-rdg 7752
This theorem is referenced by:  rdgeq12  7755  rdgsucmpt2  7772  frsucmpt2  7781  seqomlem0  7790  omv  7839  oev  7841  dffi3  8586  hsmex  9549  axdc  9638  seqeq2  13048  seqval  13055  trpredlem1  32069  trpredtr  32072  trpredmintr  32073  neibastop2  32699  dffinxpf  33557  finxpeq1  33558
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