Theorem tfrlemiubacc 6309

Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6311). (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 6307	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
7		fndm 5297	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝑥 → dom ∪ 𝐵 = 𝑥)
8	6, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
9	1, 2, 3, 4, 5	tfrlemibacc 6305	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
10	9	unissd 3820	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
11	1	recsfval 6294	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
12	10, 11	sseqtrrdi 3196	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
13		dmss 4810	. . . . . . . 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 3184	. . . . . 6 ⊢ (𝜑 → 𝑥 ⊆ dom recs(𝐹))
16	15	sselda 3147	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom recs(𝐹))
17	1	tfrlem9 6298	. . . . 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 6296	. . . . . 6 ⊢ Fun recs(𝐹)
20	19	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → Fun recs(𝐹))
21	12	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → ∪ 𝐵 ⊆ recs(𝐹))
22	8	eleq2d 2240	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝑥))
23	22	biimpar 295	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom ∪ 𝐵)
24		funssfv 5522	. . . . 5 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
25	20, 21, 23, 24	syl3anc 1233	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
26		eloni 4360	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	4, 26	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝑥)
28		ordelss 4364	. . . . . . . 8 ⊢ ((Ord 𝑥 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝑥)
29	27, 28	sylan 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝑥)
30	8	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → dom ∪ 𝐵 = 𝑥)
31	29, 30	sseqtrrd 3186	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ dom ∪ 𝐵)
32		fun2ssres 5241	. . . . . 6 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
33	20, 21, 31, 32	syl3anc 1233	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
34	33	fveq2d 5500	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
35	18, 25, 34	3eqtr3d 2211	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
36	35	ralrimiva 2543	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
37		fveq2 5496	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
38		reseq2 4886	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5500	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐹‘(∪ 𝐵 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
40	37, 39	eqeq12d 2185	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤))))
41	40	cbvralv 2696	. 2 ⊢ (∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
42	36, 41	sylibr 133	1 ⊢ (𝜑 → ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ∪ cun 3119   ⊆ wss 3121  {csn 3583  ⟨cop 3586  ∪ cuni 3796  Ord word 4347  Oncon0 4348  dom cdm 4611   ↾ cres 4613  Fun wfun 5192   Fn wfn 5193  ‘cfv 5198  recscrecs 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-recs 6284

This theorem is referenced by:  tfrlemiex  6310