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Theorem frecsuclem1 5926
Description: Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 13-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
frecsuclem1 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc 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 frecsuclem1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 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)
2 frecsuclem1.h . . . . . . . 8 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
3 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))})))
42, 3ax-mp 7 . . . . . . 7 recs(𝐺) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
54reseq1i 4551 . . . . . 6 (recs(𝐺) ↾ 𝜔) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
61, 5eqtr4i 2060 . . . . 5 frec(𝐹, A) = (recs(𝐺) ↾ 𝜔)
76fveq1i 5122 . . . 4 (frec(𝐹, A)‘suc B) = ((recs(𝐺) ↾ 𝜔)‘suc B)
8 peano2 4261 . . . . 5 (B 𝜔 → suc B 𝜔)
9 fvres 5141 . . . . 5 (suc B 𝜔 → ((recs(𝐺) ↾ 𝜔)‘suc B) = (recs(𝐺)‘suc B))
108, 9syl 14 . . . 4 (B 𝜔 → ((recs(𝐺) ↾ 𝜔)‘suc B) = (recs(𝐺)‘suc B))
117, 10syl5eq 2081 . . 3 (B 𝜔 → (frec(𝐹, A)‘suc B) = (recs(𝐺)‘suc B))
12113ad2ant3 926 . 2 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (recs(𝐺)‘suc B))
13 nnon 4275 . . . . 5 (suc B 𝜔 → suc B On)
148, 13syl 14 . . . 4 (B 𝜔 → suc B On)
15 eqid 2037 . . . . 5 recs(𝐺) = recs(𝐺)
162frectfr 5924 . . . . 5 ((z(𝐹z) V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
1715, 16tfri2d 5891 . . . 4 (((z(𝐹z) V A 𝑉) suc B On) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
1814, 17sylan2 270 . . 3 (((z(𝐹z) V A 𝑉) B 𝜔) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
19183impa 1098 . 2 ((z(𝐹z) V A 𝑉 B 𝜔) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
2012, 19eqtrd 2069 1 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc 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  Oncon0 4066  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-in1 544  ax-in2 545  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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-frec 5918
This theorem is referenced by:  frecsuclem3  5929
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