Theorem tfrlemiubacc 6474

Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6476). (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 6472	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
7		fndm 5419	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝑥 → dom ∪ 𝐵 = 𝑥)
8	6, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
9	1, 2, 3, 4, 5	tfrlemibacc 6470	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
10	9	unissd 3911	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
11	1	recsfval 6459	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
12	10, 11	sseqtrrdi 3273	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
13		dmss 4921	. . . . . . . 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 3261	. . . . . 6 ⊢ (𝜑 → 𝑥 ⊆ dom recs(𝐹))
16	15	sselda 3224	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom recs(𝐹))
17	1	tfrlem9 6463	. . . . 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 6461	. . . . . 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 2299	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝑥))
23	22	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom ∪ 𝐵)
24		funssfv 5652	. . . . 5 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
25	20, 21, 23, 24	syl3anc 1271	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
26		eloni 4465	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	4, 26	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝑥)
28		ordelss 4469	. . . . . . . 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 3263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ dom ∪ 𝐵)
32		fun2ssres 5360	. . . . . 6 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
33	20, 21, 31, 32	syl3anc 1271	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
34	33	fveq2d 5630	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
35	18, 25, 34	3eqtr3d 2270	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
36	35	ralrimiva 2603	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
37		fveq2 5626	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
38		reseq2 4999	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5630	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐹‘(∪ 𝐵 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
40	37, 39	eqeq12d 2244	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤))))
41	40	cbvralv 2765	. 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 1002  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197  {csn 3666  ⟨cop 3669  ∪ cuni 3887  Ord word 4452  Oncon0 4453  dom cdm 4718   ↾ cres 4720  Fun wfun 5311   Fn wfn 5312  ‘cfv 5317  recscrecs 6448

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-recs 6449

This theorem is referenced by:  tfrlemiex  6475