Theorem tfrcllemubacc 6630

Description: Lemma for tfrcl 6635. 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 6628	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)
11		fdm 5539	. . . . . . . 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 6626	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
14	13	unissd 3959	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
15	5, 3	tfrcllemssrecs 6623	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
16	14, 15	sstrd 3258	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
17		dmss 4980	. . . . . . . 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 3285	. . . . . 6 ⊢ (𝜑 → 𝐷 ⊆ dom recs(𝐺))
20	19	sselda 3248	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom recs(𝐺))
21		eqid 2238	. . . . . 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 6590	. . . . 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 6591	. . . . 5 ⊢ Fun recs(𝐺)
25	12	eleq2d 2308	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝐷))
26	25	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ∈ dom ∪ 𝐵)
27		funssfv 5721	. . . . 5 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
28	24, 16, 26, 27	mp3an2ani 1385	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺)‘𝑤) = (∪ 𝐵‘𝑤))
29		ordelon 4528	. . . . . . . . . 10 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
30	3, 8, 29	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ On)
31		eloni 4520	. . . . . . . . 9 ⊢ (𝐷 ∈ On → Ord 𝐷)
32	30, 31	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝐷)
33		ordelss 4524	. . . . . . . 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 3287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → 𝑤 ⊆ dom ∪ 𝐵)
37		fun2ssres 5421	. . . . . 6 ⊢ ((Fun recs(𝐺) ∧ ∪ 𝐵 ⊆ recs(𝐺) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
38	24, 16, 36, 37	mp3an2ani 1385	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (recs(𝐺) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5699	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (𝐺‘(recs(𝐺) ↾ 𝑤)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
40	23, 28, 39	3eqtr3d 2279	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
41	40	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐷 (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
42		fveq2 5695	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
43		reseq2 5058	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
44	43	fveq2d 5699	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐺‘(∪ 𝐵 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑤)))
45	42, 44	eqeq12d 2253	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐺‘(∪ 𝐵 ↾ 𝑤))))
46	45	cbvralv 2786	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529   ∪ cun 3218   ⊆ wss 3220  {csn 3709  ⟨cop 3712  ∪ cuni 3935  Ord word 4507  Oncon0 4508  suc csuc 4510  dom cdm 4774   ↾ cres 4776  Fun wfun 5371   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  recscrecs 6575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-recs 6576

This theorem is used by:  tfrcllemex  6631