Theorem tfrlemiubacc 6415

Description: The union of 𝐵 satisfies the recursion rule (lemma for tfrlemi1 6417). (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 6413	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
7		fndm 5372	. . . . . . . 8 ⊢ (∪ 𝐵 Fn 𝑥 → dom ∪ 𝐵 = 𝑥)
8	6, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
9	1, 2, 3, 4, 5	tfrlemibacc 6411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
10	9	unissd 3873	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
11	1	recsfval 6400	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
12	10, 11	sseqtrrdi 3241	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
13		dmss 4876	. . . . . . . 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 3229	. . . . . 6 ⊢ (𝜑 → 𝑥 ⊆ dom recs(𝐹))
16	15	sselda 3192	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom recs(𝐹))
17	1	tfrlem9 6404	. . . . 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 6402	. . . . . 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 2274	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝐵 ↔ 𝑤 ∈ 𝑥))
23	22	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ dom ∪ 𝐵)
24		funssfv 5601	. . . . 5 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ∈ dom ∪ 𝐵) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
25	20, 21, 23, 24	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹)‘𝑤) = (∪ 𝐵‘𝑤))
26		eloni 4421	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	4, 26	syl 14	. . . . . . . 8 ⊢ (𝜑 → Ord 𝑥)
28		ordelss 4425	. . . . . . . 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 3231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ dom ∪ 𝐵)
32		fun2ssres 5313	. . . . . 6 ⊢ ((Fun recs(𝐹) ∧ ∪ 𝐵 ⊆ recs(𝐹) ∧ 𝑤 ⊆ dom ∪ 𝐵) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
33	20, 21, 31, 32	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (recs(𝐹) ↾ 𝑤) = (∪ 𝐵 ↾ 𝑤))
34	33	fveq2d 5579	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (𝐹‘(recs(𝐹) ↾ 𝑤)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
35	18, 25, 34	3eqtr3d 2245	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑥) → (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
36	35	ralrimiva 2578	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑥 (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
37		fveq2 5575	. . . 4 ⊢ (𝑢 = 𝑤 → (∪ 𝐵‘𝑢) = (∪ 𝐵‘𝑤))
38		reseq2 4953	. . . . 5 ⊢ (𝑢 = 𝑤 → (∪ 𝐵 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑤))
39	38	fveq2d 5579	. . . 4 ⊢ (𝑢 = 𝑤 → (𝐹‘(∪ 𝐵 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑤)))
40	37, 39	eqeq12d 2219	. . 3 ⊢ (𝑢 = 𝑤 → ((∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑤) = (𝐹‘(∪ 𝐵 ↾ 𝑤))))
41	40	cbvralv 2737	. 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 980  ∀wal 1370   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∃wrex 2484  Vcvv 2771   ∪ cun 3163   ⊆ wss 3165  {csn 3632  ⟨cop 3635  ∪ cuni 3849  Ord word 4408  Oncon0 4409  dom cdm 4674   ↾ cres 4676  Fun wfun 5264   Fn wfn 5265  ‘cfv 5270  recscrecs 6389

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-recs 6390

This theorem is referenced by:  tfrlemiex  6416