Theorem frecsuclem2 5989

Description: Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis

Ref	Expression
frecsuclem1.h	⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})

Assertion

Ref	Expression
frecsuclem2	⊢ ((∀𝑧(𝐹‘𝑧) ∈ 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

Step	Hyp	Ref	Expression
1		sucidg 4153	. . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ suc 𝐵)
2		fvres 5198	. . . 4 ⊢ (𝐵 ∈ suc 𝐵 → ((recs(𝐺) ↾ suc 𝐵)‘𝐵) = (recs(𝐺)‘𝐵))
3	1, 2	syl 14	. . 3 ⊢ (𝐵 ∈ ω → ((recs(𝐺) ↾ suc 𝐵)‘𝐵) = (recs(𝐺)‘𝐵))
4		df-frec 5978	. . . . . 6 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
5		frecsuclem1.h	. . . . . . . 8 ⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
6		recseq 5921	. . . . . . . 8 ⊢ (𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) → recs(𝐺) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})))
7	5, 6	ax-mp 7	. . . . . . 7 ⊢ recs(𝐺) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
8	7	reseq1i 4608	. . . . . 6 ⊢ (recs(𝐺) ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
9	4, 8	eqtr4i 2063	. . . . 5 ⊢ frec(𝐹, 𝐴) = (recs(𝐺) ↾ ω)
10	9	fveq1i 5179	. . . 4 ⊢ (frec(𝐹, 𝐴)‘𝐵) = ((recs(𝐺) ↾ ω)‘𝐵)
11		fvres 5198	. . . 4 ⊢ (𝐵 ∈ ω → ((recs(𝐺) ↾ ω)‘𝐵) = (recs(𝐺)‘𝐵))
12	10, 11	syl5eq 2084	. . 3 ⊢ (𝐵 ∈ ω → (frec(𝐹, 𝐴)‘𝐵) = (recs(𝐺)‘𝐵))
13	3, 12	eqtr4d 2075	. 2 ⊢ (𝐵 ∈ ω → ((recs(𝐺) ↾ suc 𝐵)‘𝐵) = (frec(𝐹, 𝐴)‘𝐵))
14	13	3ad2ant3 927	1 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → ((recs(𝐺) ↾ suc 𝐵)‘𝐵) = (frec(𝐹, 𝐴)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629   ∧ w3a 885  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  Vcvv 2557  ∅c0 3224   ↦ cmpt 3818  suc csuc 4102  ωcom 4313  dom cdm 4345   ↾ cres 4347  ‘cfv 4902  recscrecs 5919  freccfrec 5977

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-suc 4108  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecsuclem3  5990