Theorem tfrlemibxssdm 6306

Description: The union of 𝐵 is defined on all ordinals. Lemma for tfrlemi1 6311. (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 6291	. . . . . . . . . . 11 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
5	4	simprd 113	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
6	5	3ad2ant1 1013	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → (𝐹‘𝑔) ∈ V)
7		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
8		opexg 4213	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
9	7, 5, 8	sylancr 412	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
10		snidg 3612	. . . . . . . . . . . 12 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
11		elun2 3295	. . . . . . . . . . . 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 1013	. . . . . . . . . 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 1019	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ 𝑥)
15		simp3l 1020	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 Fn 𝑧)
16		onelon 4369	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
17		rspe 2519	. . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ 𝑔 ∈ V
21	19, 20	tfrlem3a 6289	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
22	18, 21	sylibr 133	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 ∈ 𝐴)
23	22	3adant1 1010	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → 𝑔 ∈ 𝐴)
24	14, 15, 23	3jca 1172	. . . . . . . . . . 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 4170	. . . . . . . . . . . . . 14 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
26		unexg 4428	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
27	20, 26	mpan 422	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V ↔ ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
30	28, 29	sylib 121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
31	30	3ad2ant1 1013	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
32		simpr3 1000	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33		19.8a 1583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34		rspe 2519	. . . . . . . . . . . . . . . . 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 2281	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
37	34, 36	sylibr 133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝐵)
38	33, 37	sylan2 284	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝐵)
39	32, 38	eqeltrrd 2248	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵)
40	39	3exp2 1220	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑔 Fn 𝑧 → (𝑔 ∈ 𝐴 → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵))))
41	40	3imp 1188	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴) → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵))
42	41	exlimdv 1812	. . . . . . . . . . 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 3801	. . . . . . . . . 10 ⊢ ((⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵)
45	13, 43, 44	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵)
46		opeq2 3766	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐹‘𝑔)⟩)
47	46	eleq1d 2239	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹‘𝑔) → (⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵 ↔ ⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵))
48	47	spcegv 2818	. . . . . . . . . 10 ⊢ ((𝐹‘𝑔) ∈ V → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ ∪ 𝐵 → ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
49	7	eldm2 4809	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom ∪ 𝐵 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
50	48, 49	syl6ibr 161	. . . . . . . . 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 1200	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥)) → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
53	52	exlimdv 1812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ On ∧ 𝑧 ∈ 𝑥)) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
54	53	anassrs 398	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
55	54	ralimdva 2537	. . . 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 419	. . 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 3137	. 2 ⊢ (𝑥 ⊆ dom ∪ 𝐵 ↔ ∀𝑧 ∈ 𝑥 𝑧 ∈ dom ∪ 𝐵)
59	57, 58	sylibr 133	1 ⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)

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  Oncon0 4348  dom cdm 4611   ↾ cres 4613  Fun wfun 5192   Fn wfn 5193  ‘cfv 5198

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-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

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-tr 4088  df-iord 4351  df-on 4353  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tfrlemibfn  6307