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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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