Theorem tfrlemiubacc 6563

Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6565). (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 6561	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
7		fndm 5457	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝑥 → dom ∪ 𝐵 = 𝑥)
8	6, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
9	1, 2, 3, 4, 5	tfrlemibacc 6559	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
10	9	unissd 3940	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
11	1	recsfval 6548	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
12	10, 11	sseqtrrdi 3289	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
13		dmss 4957	. . . . . . . 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 3277	. . . . . 6 ⊢ (𝜑 → 𝑥 ⊆ dom recs(𝐹))
16	15	sselda 3240	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom recs(𝐹))
17	1	tfrlem9 6552	. . . . 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 6550	. . . . . 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 2304	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝑥))
23	22	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom ∪ 𝐵)
24		funssfv 5698	. . . . 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 4498	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	4, 26	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝑥)
28		ordelss 4502	. . . . . . . 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 3279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ dom ∪ 𝐵)
32		fun2ssres 5398	. . . . . 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 5676	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
35	18, 25, 34	3eqtr3d 2275	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
36	35	ralrimiva 2617	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
37		fveq2 5672	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
38		reseq2 5035	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5676	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐹‘(∪ 𝐵 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
40	37, 39	eqeq12d 2249	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤))))
41	40	cbvralv 2780	. 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 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ∪ cun 3211   ⊆ wss 3213  {csn 3691  ⟨cop 3694  ∪ cuni 3916  Ord word 4485  Oncon0 4486  dom cdm 4751   ↾ cres 4753  Fun wfun 5348   Fn wfn 5349  ‘cfv 5354  recscrecs 6537

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-recs 6538

This theorem is referenced by:  tfrlemiex  6564