Theorem tfrlemibxssdm 6380

Description: The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 6385. (Contributed by Jim Kingdon, 18-Mar-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
tfrlemibxssdm	⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)

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

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

Proof of Theorem tfrlemibxssdm

Step	Hyp	Ref	Expression
1		tfrlemi1.5	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
2		tfrlemi1.4	. . . 4 ⊢ (𝜑 → 𝑥 ∈ On)
3		tfrlemisucfn.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
4	3	tfrlem3-2d 6365	. . . . . . . . . . 11 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
5	4	simprd 114	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
6	5	3ad2ant1 1020	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → (𝐹‘𝑔) ∈ V)
7		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
8		opexg 4257	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
9	7, 5, 8	sylancr 414	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
10		snidg 3647	. . . . . . . . . . . 12 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
11		elun2 3327	. . . . . . . . . . . 12 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩} → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
12	9, 10, 11	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
13	12	3ad2ant1 1020	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
14		simp2r 1026	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ 𝑥)
15		simp3l 1027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 Fn 𝑧)
16		onelon 4415	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
17		rspe 2543	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
18	16, 17	sylan 283	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
19		tfrlemisucfn.1	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
20		vex 2763	. . . . . . . . . . . . . . 15 ⊢ 𝑔 ∈ V
21	19, 20	tfrlem3a 6363	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
22	18, 21	sylibr 134	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 ∈ 𝐴)
23	22	3adant1 1017	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 ∈ 𝐴)
24	14, 15, 23	3jca 1179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → (𝑧 ∈ 𝑥 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴))
25		snexg 4213	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
26		unexg 4474	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
27	20, 26	mpan 424	. . . . . . . . . . . . . 14 ⊢ ({⟨𝑧, (𝐹‘𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
28	9, 25, 27	3syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
29		isset 2766	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V ↔ ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
30	28, 29	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
31	30	3ad2ant1 1020	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
32		simpr3 1007	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33		19.8a 1601	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34		rspe 2543	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
35		tfrlemi1.3	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
36	35	abeq2i 2304	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
37	34, 36	sylibr 134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝐵)
38	33, 37	sylan2 286	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝐵)
39	32, 38	eqeltrrd 2271	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵)
40	39	3exp2 1227	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑔 Fn 𝑧 → (𝑔 ∈ 𝐴 → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵))))
41	40	3imp 1195	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴) → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵))
42	41	exlimdv 1830	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴) → (∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵))
43	24, 31, 42	sylc 62	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵)
44		elunii 3840	. . . . . . . . . 10 ⊢ ((⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵)
45	13, 43, 44	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵)
46		opeq2 3805	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐹‘𝑔)⟩)
47	46	eleq1d 2262	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹‘𝑔) → (⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵 ↔ ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵))
48	47	spcegv 2848	. . . . . . . . . 10 ⊢ ((𝐹‘𝑔) ∈ V → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵 → ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
49	7	eldm2 4860	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom ∪ 𝐵 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
50	48, 49	imbitrrdi 162	. . . . . . . . 9 ⊢ ((𝐹‘𝑔) ∈ V → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵 → 𝑧 ∈ dom ∪ 𝐵))
51	6, 45, 50	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ dom ∪ 𝐵)
52	51	3expia 1207	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥)) → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
53	52	exlimdv 1830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥)) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
54	53	anassrs 400	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
55	54	ralimdva 2561	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → (∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → ∀𝑧 ∈ 𝑥 𝑧 ∈ dom ∪ 𝐵))
56	2, 55	mpdan 421	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → ∀𝑧 ∈ 𝑥 𝑧 ∈ dom ∪ 𝐵))
57	1, 56	mpd 13	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 𝑧 ∈ dom ∪ 𝐵)
58		dfss3 3169	. 2 ⊢ (𝑥 ⊆ dom ∪ 𝐵 ↔ ∀𝑧 ∈ 𝑥 𝑧 ∈ dom ∪ 𝐵)
59	57, 58	sylibr 134	1 ⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153  {csn 3618  ⟨cop 3621  ∪ cuni 3835  Oncon0 4394  dom cdm 4659   ↾ cres 4661  Fun wfun 5248   Fn wfn 5249  ‘cfv 5254

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-tr 4128  df-iord 4397  df-on 4399  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  tfrlemibfn  6381