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Theorem rdgeq1 7660
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 6329 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(𝑔 dom 𝑔)) = (𝐺‘(𝑔 dom 𝑔)))
21ifeq2d 4244 . . . . 5 (𝐹 = 𝐺 → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))
32ifeq2d 4244 . . . 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 4879 . . 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 7623 . . 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 7659 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
8 df-rdg 7659 . 2 rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
96, 7, 83eqtr4g 2830 1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  Vcvv 3351  c0 4063  ifcif 4225   cuni 4574  cmpt 4863  dom cdm 5249  ran crn 5250  Lim wlim 5865  cfv 6029  recscrecs 7620  reccrdg 7658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-iota 5992  df-fv 6037  df-wrecs 7559  df-recs 7621  df-rdg 7659
This theorem is referenced by:  rdgeq12  7662  rdgsucmpt2  7679  frsucmpt2  7688  seqomlem0  7697  omv  7746  oev  7748  dffi3  8493  hsmex  9456  axdc  9545  seqeq2  13008  seqval  13015  trpredlem1  32059  trpredtr  32062  trpredmintr  32063  neibastop2  32689  dffinxpf  33555  finxpeq1  33556
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