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Theorem freceq1 5919
 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 102 . . . . . . . . . . 11 ((𝐹 = 𝐺 g V) → 𝐹 = 𝐺)
21fveq1d 5123 . . . . . . . . . 10 ((𝐹 = 𝐺 g V) → (𝐹‘(g𝑚)) = (𝐺‘(g𝑚)))
32eleq2d 2104 . . . . . . . . 9 ((𝐹 = 𝐺 g V) → (x (𝐹‘(g𝑚)) ↔ x (𝐺‘(g𝑚))))
43anbi2d 437 . . . . . . . 8 ((𝐹 = 𝐺 g V) → ((dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ (dom g = suc 𝑚 x (𝐺‘(g𝑚)))))
54rexbidv 2321 . . . . . . 7 ((𝐹 = 𝐺 g V) → (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ 𝑚 𝜔 (dom g = suc 𝑚 x (𝐺‘(g𝑚)))))
65orbi1d 704 . . . . . 6 ((𝐹 = 𝐺 g V) → ((𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A)) ↔ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐺‘(g𝑚))) (dom g = ∅ x A))))
76abbidv 2152 . . . . 5 ((𝐹 = 𝐺 g V) → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐺‘(g𝑚))) (dom g = ∅ x A))})
87mpteq2dva 3838 . . . 4 (𝐹 = 𝐺 → (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 (𝐺‘(g𝑚))) (dom g = ∅ x A))}))
9 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 (𝐺‘(g𝑚))) (dom g = ∅ x A))}) → 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 (𝐺‘(g𝑚))) (dom g = ∅ x A))})))
108, 9syl 14 . . 3 (𝐹 = 𝐺 → 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 (𝐺‘(g𝑚))) (dom g = ∅ x A))})))
1110reseq1d 4554 . 2 (𝐹 = 𝐺 → (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 (𝐺‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔))
12 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)
13 df-frec 5918 . 2 frec(𝐺, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐺‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
1411, 12, 133eqtr4g 2094 1 (𝐹 = 𝐺 → frec(𝐹, A) = frec(𝐺, A))
 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  iseqeq2  8895  iseqeq3  8896  iseqeq4  8897  iseqval  8900
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