Theorem tfr1onlemubacc 6577

Description: Lemma for tfr1on 6581. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	⊢ 𝐹 = recs(𝐺)
tfr1on.g	⊢ (𝜑 → Fun 𝐺)
tfr1on.x	⊢ (𝜑 → Ord 𝑋)
tfr1on.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
tfr1onlemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfr1onlembacc.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
tfr1onlembacc.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfr1onlembacc.4	⊢ (𝜑 → 𝐷 ∈ 𝑋)
tfr1onlembacc.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfr1onlemubacc	⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))

Distinct variable groups:   𝐴,𝑓,𝑔,ℎ,𝑥,𝑧   𝐷,𝑓,𝑔,𝑥   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,ℎ,𝑥,𝑧   𝑦,𝑔,𝑧   𝐵,𝑔,ℎ,𝑤,𝑧   𝑢,𝐵,𝑤   𝐷,ℎ,𝑤,𝑧,𝑓,𝑥   𝑢,𝐷   ℎ,𝐺,𝑧,𝑦   𝑢,𝐺,𝑤   𝑔,𝑋,𝑧   𝜑,𝑤   𝑦,𝑤

Allowed substitution hints:   𝜑(𝑦,𝑢)   𝐴(𝑦,𝑤,𝑢)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑦)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓,𝑔,ℎ)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑢,ℎ)

Proof of Theorem tfr1onlemubacc

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . . . . . . 9 ⊢ 𝐹 = recs(𝐺)
2		tfr1on.g	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
3		tfr1on.x	. . . . . . . . 9 ⊢ (𝜑 → Ord 𝑋)
4		tfr1on.ex	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
5		tfr1onlemsucfn.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfr1onlembacc.3	. . . . . . . . 9 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfr1onlembacc.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfr1onlembacc.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfr1onlembacc.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembfn 6575	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)
11		fndm 5455	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝐷 → dom ∪ 𝐵 = 𝐷)
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝐷)
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembacc 6573	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
14	13	unissd 3938	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
15	5, 3	tfr1onlemssrecs 6570	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
16	14, 15	sstrd 3248	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
17		dmss 4955	. . . . . . . 8 ⊢ (∪ 𝐵 ⊆ recs(𝐺) → dom ∪ 𝐵 ⊆ dom recs(𝐺))
18	16, 17	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom recs(𝐺))
19	12, 18	eqsstrrd 3275	. . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ dom recs(𝐺))
20	19	sselda 3238	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom recs(𝐺))
21		eqid 2232	. . . . . 6 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
22	21	tfrlem9 6550	. . . . 5 ⊢ (𝑤 ∈ dom recs(𝐺) → (recs(𝐺)‘𝑤) = (𝐺‘(recs(𝐺) ↾ 𝑤)))
23	20, 22	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺)‘𝑤) = (𝐺‘(recs(𝐺) ↾ 𝑤)))
24		tfrfun 6551	. . . . 5 ⊢ Fun recs(𝐺)
25	12	eleq2d 2302	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝐷))
26	25	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom ∪ 𝐵)
27		funssfv 5696	. . . . 5 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
28	24, 16, 26, 27	mp3an2ani 1381	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
29		ordelon 4504	. . . . . . . . . 10 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
30	3, 8, 29	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ On)
31		eloni 4496	. . . . . . . . 9 ⊢ (𝐷 ∈ On → Ord 𝐷)
32	30, 31	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝐷)
33		ordelss 4500	. . . . . . . 8 ⊢ ((Ord 𝐷 ∧ 𝑤 ∈ 𝐷) → 𝑤 ⊆ 𝐷)
34	32, 33	sylan 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ⊆ 𝐷)
35	12	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → dom ∪ 𝐵 = 𝐷)
36	34, 35	sseqtrrd 3277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ⊆ dom ∪ 𝐵)
37		fun2ssres 5396	. . . . . 6 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
38	24, 16, 36, 37	mp3an2ani 1381	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5674	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
40	23, 28, 39	3eqtr3d 2273	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
41	40	ralrimiva 2615	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
42		fveq2 5670	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
43		reseq2 5033	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
44	43	fveq2d 5674	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘(∪ 𝐵 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
45	42, 44	eqeq12d 2247	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤))))
46	45	cbvralv 2778	. 2 ⊢ (∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)) ↔ ∀𝑤 ∈ 𝐷 (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
47	41, 46	sylibr 134	1 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ∪ cun 3209   ⊆ wss 3211  {csn 3689  ⟨cop 3692  ∪ cuni 3914  Ord word 4483  Oncon0 4484  suc csuc 4486  dom cdm 4749   ↾ cres 4751  Fun wfun 5346   Fn wfn 5347  ‘cfv 5352  recscrecs 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-recs 6536

This theorem is referenced by:  tfr1onlemex  6578