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Theorem rdgeq1 7504
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 6188 . . . . . 6 (𝐹 = 𝐺 → (𝐹‘(𝑔 dom 𝑔)) = (𝐺‘(𝑔 dom 𝑔)))
21ifeq2d 4103 . . . . 5 (𝐹 = 𝐺 → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))
32ifeq2d 4103 . . . 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 4743 . . 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 7467 . . 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 7503 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
8 df-rdg 7503 . 2 rec(𝐺, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐺‘(𝑔 dom 𝑔))))))
96, 7, 83eqtr4g 2680 1 (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  Vcvv 3198  c0 3913  ifcif 4084   cuni 4434  cmpt 4727  dom cdm 5112  ran crn 5113  Lim wlim 5722  cfv 5886  recscrecs 7464  reccrdg 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-iota 5849  df-fv 5894  df-wrecs 7404  df-recs 7465  df-rdg 7503
This theorem is referenced by:  rdgeq12  7506  rdgsucmpt2  7523  frsucmpt2  7532  seqomlem0  7541  omv  7589  oev  7591  dffi3  8334  hsmex  9251  axdc  9340  seqeq2  12800  seqval  12807  trpredlem1  31711  trpredtr  31714  trpredmintr  31715  neibastop2  32340  dffinxpf  33202  finxpeq1  33203
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