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Theorem freceq2 5920
 Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
freceq2 (A = B → frec(𝐹, A) = frec(𝐹, B))

Proof of Theorem freceq2
Dummy variables x g 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . . . . 9 ((A = B g V) → A = B)
21eleq2d 2104 . . . . . . . 8 ((A = B g V) → (x Ax B))
32anbi2d 437 . . . . . . 7 ((A = B g V) → ((dom g = ∅ x A) ↔ (dom g = ∅ x B)))
43orbi2d 703 . . . . . 6 ((A = B g V) → ((𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A)) ↔ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))))
54abbidv 2152 . . . . 5 ((A = B g V) → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))})
65mpteq2dva 3838 . . . 4 (A = B → (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))}))
7 recseq 5862 . . . 4 ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))}) → recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))})))
86, 7syl 14 . . 3 (A = B → recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))})))
98reseq1d 4554 . 2 (A = B → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))})) ↾ 𝜔))
10 df-frec 5918 . 2 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
11 df-frec 5918 . 2 frec(𝐹, B) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x B))})) ↾ 𝜔)
129, 10, 113eqtr4g 2094 1 (A = B → frec(𝐹, A) = frec(𝐹, B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  ∅c0 3218   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288   ↾ cres 4290  ‘cfv 4845  recscrecs 5860  freccfrec 5917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-res 4300  df-iota 4810  df-fv 4853  df-recs 5861  df-frec 5918 This theorem is referenced by:  iseqeq1  8894  iseqeq3  8896
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