Theorem tmdgsum2 24038

Description: For any neighborhood 𝑈 of 𝑛𝑋, there is a neighborhood 𝑢 of 𝑋 such that any sum of 𝑛 elements in 𝑢 sums to an element of 𝑈. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
tmdgsum.j	⊢ 𝐽 = (TopOpen‘𝐺)
tmdgsum.b	⊢ 𝐵 = (Base‘𝐺)
tmdgsum2.t	⊢ · = (.g‘𝐺)
tmdgsum2.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tmdgsum2.2	⊢ (𝜑 → 𝐺 ∈ TopMnd)
tmdgsum2.a	⊢ (𝜑 → 𝐴 ∈ Fin)
tmdgsum2.u	⊢ (𝜑 → 𝑈 ∈ 𝐽)
tmdgsum2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
tmdgsum2.3	⊢ (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)

Assertion

Ref	Expression
tmdgsum2	⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))

Distinct variable groups:   𝑢,𝑓,𝐴   𝑓,𝐽,𝑢   𝑓,𝑋,𝑢   𝐵,𝑓,𝑢   𝑓,𝐺,𝑢   𝑈,𝑓,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑓)   · (𝑢,𝑓)

Proof of Theorem tmdgsum2

Dummy variables 𝑔 𝑘 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓))
2	1	mptpreima 6194	. . . . . 6 ⊢ (◡(𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}
3		tmdgsum2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd)
4		tmdgsum2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
5		tmdgsum2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		tmdgsum.j	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺)
7		tmdgsum.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
8	6, 7	tmdgsum 24037	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽))
9	3, 4, 5, 8	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽))
10		tmdgsum2.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
11		cnima 23207	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽) ∧ 𝑈 ∈ 𝐽) → (◡(𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽 ↑ko 𝒫 𝐴))
12	9, 10, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (◡(𝑓 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽 ↑ko 𝒫 𝐴))
13	2, 12	eqeltrrid 2839	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽 ↑ko 𝒫 𝐴))
14	6, 7	tmdtopon 24023	. . . . . . . 8 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15		topontop 22855	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
16	4, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
17		xkopt 23597	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
18	16, 5, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐽 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
19		fnconstg 6720	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
20	4, 14, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21		eqid 2734	. . . . . . . 8 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
22	21	ptval 23512	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 × {𝐽}) Fn 𝐴) → (∏t‘(𝐴 × {𝐽})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
23	5, 20, 22	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (∏t‘(𝐴 × {𝐽})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
24	18, 23	eqtrd 2769	. . . . 5 ⊢ (𝜑 → (𝐽 ↑ko 𝒫 𝐴) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
25	13, 24	eleqtrd 2836	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
26		oveq2 7364	. . . . . 6 ⊢ (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
27	26	eleq1d 2819	. . . . 5 ⊢ (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
28		tmdgsum2.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
29		fconst6g 6721	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (𝐴 × {𝑋}):𝐴⟶𝐵)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 × {𝑋}):𝐴⟶𝐵)
31	7	fvexi 6846	. . . . . . 7 ⊢ 𝐵 ∈ V
32		elmapg 8774	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
33	31, 5, 32	sylancr 587	. . . . . 6 ⊢ (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵 ↑m 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
34	30, 33	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵 ↑m 𝐴))
35		fconstmpt 5684	. . . . . . . 8 ⊢ (𝐴 × {𝑋}) = (𝑘 ∈ 𝐴 ↦ 𝑋)
36	35	oveq2i 7367	. . . . . . 7 ⊢ (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))
37		cmnmnd 19724	. . . . . . . . 9 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
38	3, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
39		tmdgsum2.t	. . . . . . . . 9 ⊢ · = (.g‘𝐺)
40	7, 39	gsumconst 19861	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
41	38, 5, 28, 40	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
42	36, 41	eqtrid 2781	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
43		tmdgsum2.3	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
44	42, 43	eqeltrd 2834	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
45	27, 34, 44	elrabd 3646	. . . 4 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
46		tg2 22907	. . . 4 ⊢ (({𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}) ∧ (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡 ∧ 𝑡 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
47	25, 45, 46	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡 ∧ 𝑡 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
48		eleq2 2823	. . . . 5 ⊢ (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
49		sseq1 3957	. . . . 5 ⊢ (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
50	48, 49	anbi12d 632	. . . 4 ⊢ (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡 ∧ 𝑡 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
51	50	rexab2 3655	. . 3 ⊢ (∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡 ∧ 𝑡 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
52	47, 51	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
53		toponuni 22856	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
54	4, 14, 53	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
55	54	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐵 = ∪ 𝐽)
56	55	ineq1d 4169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐵 ∩ ∩ ran 𝑔) = (∪ 𝐽 ∩ ∩ ran 𝑔))
57	16	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐽 ∈ Top)
58		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔 Fn 𝐴)
59		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))
60		fvconst2g 7146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
61	60	eleq2d 2820	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔‘𝑦) ∈ 𝐽))
62	61	ralbidva 3155	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
63	57, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
64	59, 63	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽)
65		ffnfv 7062	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
66	58, 64, 65	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔:𝐴⟶𝐽)
67	66	frnd 6668	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ran 𝑔 ⊆ 𝐽)
68	5	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐴 ∈ Fin)
69		dffn4 6750	. . . . . . . . . . . . . 14 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
70	58, 69	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔:𝐴–onto→ran 𝑔)
71		fofi 9211	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑔:𝐴–onto→ran 𝑔) → ran 𝑔 ∈ Fin)
72	68, 70, 71	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ran 𝑔 ∈ Fin)
73		eqid 2734	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	rintopn 22851	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ran 𝑔 ⊆ 𝐽 ∧ ran 𝑔 ∈ Fin) → (∪ 𝐽 ∩ ∩ ran 𝑔) ∈ 𝐽)
75	57, 67, 72, 74	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (∪ 𝐽 ∩ ∩ ran 𝑔) ∈ 𝐽)
76	56, 75	eqeltrd 2834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐵 ∩ ∩ ran 𝑔) ∈ 𝐽)
77	28	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑋 ∈ 𝐵)
78		fconstmpt 5684	. . . . . . . . . . . . . 14 ⊢ (𝐴 × {𝑋}) = (𝑦 ∈ 𝐴 ↦ 𝑋)
79		simprl 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
80	78, 79	eqeltrrid 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
81		mptelixpg 8871	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → ((𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
82	68, 81	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ((𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
83	80, 82	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦))
84		eleq2 2823	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝑔‘𝑦)))
85	84	ralrn 7031	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
86	58, 85	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
87	83, 86	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧)
88		elrint 4942	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧))
89	77, 87, 88	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔))
90	31	inex1 5260	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ ∩ ran 𝑔) ∈ V
91		ixpconstg 8842	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∩ ∩ ran 𝑔) ∈ V) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) = ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴))
92	68, 90, 91	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) = ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴))
93		inss2 4188	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
94		fnfvelrn 7023	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ran 𝑔)
95		intss1 4916	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘𝑦) ∈ ran 𝑔 → ∩ ran 𝑔 ⊆ (𝑔‘𝑦))
96	94, 95	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ∩ ran 𝑔 ⊆ (𝑔‘𝑦))
97	93, 96	sstrid 3943	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
98	97	ralrimiva 3126	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → ∀𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
99		ss2ixp 8846	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
100	58, 98, 99	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
101	92, 100	eqsstrrd 3967	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
102		ssrab 4021	. . . . . . . . . . . . 13 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ (𝐵 ↑m 𝐴) ∧ ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σg 𝑓) ∈ 𝑈))
103	102	simprbi 496	. . . . . . . . . . . 12 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} → ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
104	103	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
105		ssralv 4000	. . . . . . . . . . 11 ⊢ (((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σg 𝑓) ∈ 𝑈 → ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
106	101, 104, 105	sylc 65	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)
107		eleq2 2823	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑋 ∈ 𝑢 ↔ 𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔)))
108		oveq1 7363	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑢 ↑m 𝐴) = ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴))
109	108	raleqdv 3294	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
110	107, 109	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → ((𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈) ↔ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
111	110	rspcev 3574	. . . . . . . . . 10 ⊢ (((𝐵 ∩ ∩ ran 𝑔) ∈ 𝐽 ∧ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
112	76, 89, 106, 111	syl12anc 836	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
113	112	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) → (((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
114	113	3adantr3 1172	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦))) → (((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
115		eleq2 2823	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
116		sseq1 3957	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
117	115, 116	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
118	117	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)) ↔ (((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
119	114, 118	syl5ibrcom 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦))) → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
120	119	expimpd 453	. . . . 5 ⊢ (𝜑 → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → (((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
121	120	exlimdv 1934	. . . 4 ⊢ (𝜑 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → (((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
122	121	impd 410	. . 3 ⊢ (𝜑 → ((∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
123	122	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
124	52, 123	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2712  ∀wral 3049  ∃wrex 3058  {crab 3397  Vcvv 3438   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4552  {csn 4578  ∪ cuni 4861  ∩ cint 4900   ↦ cmpt 5177   × cxp 5620  ^◡ccnv 5621  ran crn 5623   “ cima 5625   Fn wfn 6485  ⟶wf 6486  –onto→wfo 6488  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  Xcixp 8833  Fincfn 8881  ♯chash 14251  Basecbs 17134  TopOpenctopn 17339  topGenctg 17355  ∏_tcpt 17356   Σ_g cgsu 17358  Mndcmnd 18657  ._gcmg 18995  CMndccmn 19707  Topctop 22835  TopOnctopon 22852   Cn ccn 23166   ↑_ko cxko 23503  TopMndctmd 24012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-fi 9312  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-rest 17340  df-0g 17359  df-gsum 17360  df-topgen 17361  df-pt 17362  df-mre 17503  df-mrc 17504  df-acs 17506  df-plusf 18562  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18707  df-mulg 18996  df-cntz 19244  df-cmn 19709  df-top 22836  df-topon 22853  df-topsp 22875  df-bases 22888  df-cn 23169  df-cnp 23170  df-cmp 23329  df-tx 23504  df-xko 23505  df-tmd 24014

This theorem is referenced by:  tsmsxp  24097