Theorem tfrlemiubacc 6539

Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6541). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemisucfn.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
tfrlemi1.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
tfrlemi1.4	⊢ (𝜑 → 𝑥 ∈ On)
tfrlemi1.5	⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfrlemiubacc	⊢ (𝜑 → ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))

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

Allowed substitution hints:   𝜑(𝑥,𝑢,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemiubacc

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		tfrlemisucfn.2	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
3		tfrlemi1.3	. . . . . . . . 9 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
4		tfrlemi1.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ On)
5		tfrlemi1.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
6	1, 2, 3, 4, 5	tfrlemibfn 6537	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
7		fndm 5436	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝑥 → dom ∪ 𝐵 = 𝑥)
8	6, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
9	1, 2, 3, 4, 5	tfrlemibacc 6535	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
10	9	unissd 3922	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
11	1	recsfval 6524	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
12	10, 11	sseqtrrdi 3277	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
13		dmss 4936	. . . . . . . 8 ⊢ (∪ 𝐵 ⊆ recs(𝐹) → dom ∪ 𝐵 ⊆ dom recs(𝐹))
14	12, 13	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom recs(𝐹))
15	8, 14	eqsstrrd 3265	. . . . . 6 ⊢ (𝜑 → 𝑥 ⊆ dom recs(𝐹))
16	15	sselda 3228	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom recs(𝐹))
17	1	tfrlem9 6528	. . . . 5 ⊢ (𝑤 ∈ dom recs(𝐹) → (recs(𝐹)‘𝑤) = (𝐹‘(recs(𝐹) ↾ 𝑤)))
18	16, 17	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹)‘𝑤) = (𝐹‘(recs(𝐹) ↾ 𝑤)))
19	1	tfrlem7 6526	. . . . . 6 ⊢ Fun recs(𝐹)
20	19	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → Fun recs(𝐹))
21	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → ∪ 𝐵 ⊆ recs(𝐹))
22	8	eleq2d 2301	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝑥))
23	22	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom ∪ 𝐵)
24		funssfv 5674	. . . . 5 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
25	20, 21, 23, 24	syl3anc 1274	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
26		eloni 4478	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	4, 26	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝑥)
28		ordelss 4482	. . . . . . . 8 ⊢ ((Ord 𝑥 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝑥)
29	27, 28	sylan 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝑥)
30	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → dom ∪ 𝐵 = 𝑥)
31	29, 30	sseqtrrd 3267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ dom ∪ 𝐵)
32		fun2ssres 5377	. . . . . 6 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
33	20, 21, 31, 32	syl3anc 1274	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
34	33	fveq2d 5652	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
35	18, 25, 34	3eqtr3d 2272	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
36	35	ralrimiva 2606	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
37		fveq2 5648	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
38		reseq2 5014	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5652	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐹‘(∪ 𝐵 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
40	37, 39	eqeq12d 2246	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤))))
41	40	cbvralv 2768	. 2 ⊢ (∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
42	36, 41	sylibr 134	1 ⊢ (𝜑 → ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  {csn 3673  ⟨cop 3676  ∪ cuni 3898  Ord word 4465  Oncon0 4466  dom cdm 4731   ↾ cres 4733  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333  recscrecs 6513

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfrlemiex  6540