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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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