Theorem tfrcllemubacc 6468

Description: Lemma for tfrcl 6473. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem tfrcllemubacc

Dummy variables 𝑒 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrcl.f	. . . . . . . . 9 ⊢ 𝐹 = recs(𝐺)
2		tfrcl.g	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
3		tfrcl.x	. . . . . . . . 9 ⊢ (𝜑 → Ord 𝑋)
4		tfrcl.ex	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
5		tfrcllemsucfn.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfrcllembacc.3	. . . . . . . . 9 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfrcllembacc.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfrcllembacc.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfrcllembacc.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembfn 6466	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)
11		fdm 5451	. . . . . . . 8 ⊢ (∪ 𝐵:𝐷⟶𝑆 → dom ∪ 𝐵 = 𝐷)
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝐷)
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembacc 6464	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
14	13	unissd 3888	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
15	5, 3	tfrcllemssrecs 6461	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
16	14, 15	sstrd 3211	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
17		dmss 4896	. . . . . . . 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 3238	. . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ dom recs(𝐺))
20	19	sselda 3201	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom recs(𝐺))
21		eqid 2207	. . . . . 6 ⊢ {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡 ∈ 𝑣 (𝑒‘𝑡) = (𝐺‘(𝑒 ↾ 𝑡)))} = {𝑒 ∣ ∃𝑣 ∈ On (𝑒 Fn 𝑣 ∧ ∀𝑡 ∈ 𝑣 (𝑒‘𝑡) = (𝐺‘(𝑒 ↾ 𝑡)))}
22	21	tfrlem9 6428	. . . . 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 6429	. . . . 5 ⊢ Fun recs(𝐺)
25	12	eleq2d 2277	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝐷))
26	25	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom ∪ 𝐵)
27		funssfv 5625	. . . . 5 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
28	24, 16, 26, 27	mp3an2ani 1357	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
29		ordelon 4448	. . . . . . . . . 10 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
30	3, 8, 29	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ On)
31		eloni 4440	. . . . . . . . 9 ⊢ (𝐷 ∈ On → Ord 𝐷)
32	30, 31	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝐷)
33		ordelss 4444	. . . . . . . 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 3240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ⊆ dom ∪ 𝐵)
37		fun2ssres 5333	. . . . . 6 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
38	24, 16, 36, 37	mp3an2ani 1357	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5603	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
40	23, 28, 39	3eqtr3d 2248	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
41	40	ralrimiva 2581	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
42		fveq2 5599	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
43		reseq2 4973	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
44	43	fveq2d 5603	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘(∪ 𝐵 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
45	42, 44	eqeq12d 2222	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤))))
46	45	cbvralv 2742	. 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 981   = wceq 1373  ∃wex 1516   ∈ wcel 2178  {cab 2193  ∀wral 2486  ∃wrex 2487   ∪ cun 3172   ⊆ wss 3174  {csn 3643  ⟨cop 3646  ∪ cuni 3864  Ord word 4427  Oncon0 4428  suc csuc 4430  dom cdm 4693   ↾ cres 4695  Fun wfun 5284   Fn wfn 5285  ⟶wf 5286  ‘cfv 5290  recscrecs 6413

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414

This theorem is referenced by:  tfrcllemex  6469