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Theorem frecsuclem2 5928
 Description: Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.)
Hypothesis
Ref Expression
frecsuclem1.h 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
Assertion
Ref Expression
frecsuclem2 ((z(𝐹z) V A 𝑉 B 𝜔) → ((recs(𝐺) ↾ suc B)‘B) = (frec(𝐹, A)‘B))
Distinct variable groups:   A,g,𝑚,x,z   B,g,𝑚,x,z   g,𝐹,𝑚,x,z   g,𝐺,𝑚,x,z   g,𝑉,𝑚,x
Allowed substitution hint:   𝑉(z)

Proof of Theorem frecsuclem2
StepHypRef Expression
1 sucidg 4119 . . . 4 (B 𝜔 → B suc B)
2 fvres 5141 . . . 4 (B suc B → ((recs(𝐺) ↾ suc B)‘B) = (recs(𝐺)‘B))
31, 2syl 14 . . 3 (B 𝜔 → ((recs(𝐺) ↾ suc B)‘B) = (recs(𝐺)‘B))
4 df-frec 5918 . . . . . 6 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
5 frecsuclem1.h . . . . . . . 8 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
6 recseq 5862 . . . . . . . 8 (𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) → recs(𝐺) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})))
75, 6ax-mp 7 . . . . . . 7 recs(𝐺) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
87reseq1i 4551 . . . . . 6 (recs(𝐺) ↾ 𝜔) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
94, 8eqtr4i 2060 . . . . 5 frec(𝐹, A) = (recs(𝐺) ↾ 𝜔)
109fveq1i 5122 . . . 4 (frec(𝐹, A)‘B) = ((recs(𝐺) ↾ 𝜔)‘B)
11 fvres 5141 . . . 4 (B 𝜔 → ((recs(𝐺) ↾ 𝜔)‘B) = (recs(𝐺)‘B))
1210, 11syl5eq 2081 . . 3 (B 𝜔 → (frec(𝐹, A)‘B) = (recs(𝐺)‘B))
133, 12eqtr4d 2072 . 2 (B 𝜔 → ((recs(𝐺) ↾ suc B)‘B) = (frec(𝐹, A)‘B))
14133ad2ant3 926 1 ((z(𝐹z) V A 𝑉 B 𝜔) → ((recs(𝐺) ↾ suc B)‘B) = (frec(𝐹, A)‘B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884  ∀wal 1240   = 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-suc 4074  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-recs 5861  df-frec 5918 This theorem is referenced by:  frecsuclem3  5929
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