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Theorem freceq1 6295
 Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
freceq1 (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))

Proof of Theorem freceq1
Dummy variables 𝑥 𝑔 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . . . . . . . . . 11 ((𝐹 = 𝐺𝑔 ∈ V) → 𝐹 = 𝐺)
21fveq1d 5429 . . . . . . . . . 10 ((𝐹 = 𝐺𝑔 ∈ V) → (𝐹‘(𝑔𝑚)) = (𝐺‘(𝑔𝑚)))
32eleq2d 2210 . . . . . . . . 9 ((𝐹 = 𝐺𝑔 ∈ V) → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐺‘(𝑔𝑚))))
43anbi2d 460 . . . . . . . 8 ((𝐹 = 𝐺𝑔 ∈ V) → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚)))))
54rexbidv 2439 . . . . . . 7 ((𝐹 = 𝐺𝑔 ∈ V) → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚)))))
65orbi1d 781 . . . . . 6 ((𝐹 = 𝐺𝑔 ∈ V) → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))))
76abbidv 2258 . . . . 5 ((𝐹 = 𝐺𝑔 ∈ V) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
87mpteq2dva 4024 . . . 4 (𝐹 = 𝐺 → (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
9 recseq 6209 . . . 4 ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) → recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})))
108, 9syl 14 . . 3 (𝐹 = 𝐺 → recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})))
1110reseq1d 4824 . 2 (𝐹 = 𝐺 → (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω))
12 df-frec 6294 . 2 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
13 df-frec 6294 . 2 frec(𝐺, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐺‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
1411, 12, 133eqtr4g 2198 1 (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  {cab 2126  ∃wrex 2418  Vcvv 2689  ∅c0 3366   ↦ cmpt 3995  suc csuc 4293  ωcom 4510  dom cdm 4545   ↾ cres 4547  ‘cfv 5129  recscrecs 6207  freccfrec 6293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3080  df-uni 3743  df-br 3936  df-opab 3996  df-mpt 3997  df-res 4557  df-iota 5094  df-fv 5137  df-recs 6208  df-frec 6294 This theorem is referenced by:  frecuzrdgdom  10220  frecuzrdgfun  10222  frecuzrdgsuct  10226  seqeq1  10250  seqeq2  10251  seqeq3  10252  iseqvalcbv  10259  hashfz1  10559  ennnfonelemr  11965  ctinfom  11970  isomninn  13394  iswomninn  13411  ismkvnn  13413
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