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Theorem tmdgsum2 24028

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 Σ_g 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 2728	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) = (𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓))
2	1	mptpreima 6247	. . . . . 6 ⊢ (^◡(𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) “ 𝑈) = {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 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 24027	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) ∈ ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽))
9	3, 4, 5, 8	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) ∈ ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽))
10		tmdgsum2.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
11		cnima 23197	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) ∈ ((𝐽 ↑_ko 𝒫 𝐴) Cn 𝐽) ∧ 𝑈 ∈ 𝐽) → (^◡(𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) “ 𝑈) ∈ (𝐽 ↑_ko 𝒫 𝐴))
12	9, 10, 11	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (^◡(𝑓 ∈ (𝐵 ↑_m 𝐴) ↦ (𝐺 Σ_g 𝑓)) “ 𝑈) ∈ (𝐽 ↑_ko 𝒫 𝐴))
13	2, 12	eqeltrrid 2834	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈} ∈ (𝐽 ↑_ko 𝒫 𝐴))
14	6, 7	tmdtopon 24013	. . . . . . . 8 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15		topontop 22843	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
16	4, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
17		xkopt 23587	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
18	16, 5, 17	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝐽 ↑_ko 𝒫 𝐴) = (∏_t‘(𝐴 × {𝐽})))
19		fnconstg 6790	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
20	4, 14, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21		eqid 2728	. . . . . . . 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 23502	. . . . . . 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 582	. . . . . 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 2768	. . . . 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 2831	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 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 7434	. . . . . 6 ⊢ (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σ_g 𝑓) = (𝐺 Σ_g (𝐴 × {𝑋})))
27	26	eleq1d 2814	. . . . 5 ⊢ (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σ_g 𝑓) ∈ 𝑈 ↔ (𝐺 Σ_g (𝐴 × {𝑋})) ∈ 𝑈))
28		tmdgsum2.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
29		fconst6g 6791	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (𝐴 × {𝑋}):𝐴⟶𝐵)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 × {𝑋}):𝐴⟶𝐵)
31	7	fvexi 6916	. . . . . . 7 ⊢ 𝐵 ∈ V
32		elmapg 8866	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵 ↑_m 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
33	31, 5, 32	sylancr 585	. . . . . 6 ⊢ (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵 ↑_m 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
34	30, 33	mpbird 256	. . . . 5 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵 ↑_m 𝐴))
35		fconstmpt 5744	. . . . . . . 8 ⊢ (𝐴 × {𝑋}) = (𝑘 ∈ 𝐴 ↦ 𝑋)
36	35	oveq2i 7437	. . . . . . 7 ⊢ (𝐺 Σ_g (𝐴 × {𝑋})) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋))
37		cmnmnd 19766	. . . . . . . . 9 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
38	3, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
39		tmdgsum2.t	. . . . . . . . 9 ⊢ · = (._g‘𝐺)
40	7, 39	gsumconst 19903	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
41	38, 5, 28, 40	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
42	36, 41	eqtrid 2780	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
43		tmdgsum2.3	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
44	42, 43	eqeltrd 2829	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝐴 × {𝑋})) ∈ 𝑈)
45	27, 34, 44	elrabd 3686	. . . 4 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈})
46		tg2 22896	. . . 4 ⊢ (({𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 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 Σ_g 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 Σ_g 𝑓) ∈ 𝑈}))
47	25, 45, 46	syl2anc 582	. . 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 Σ_g 𝑓) ∈ 𝑈}))
48		eleq2 2818	. . . . 5 ⊢ (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
49		sseq1 4007	. . . . 5 ⊢ (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}))
50	48, 49	anbi12d 630	. . . 4 ⊢ (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡 ∧ 𝑡 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈})))
51	50	rexab2 3696	. . 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 Σ_g 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 Σ_g 𝑓) ∈ 𝑈})))
52	47, 51	sylib 217	. 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 Σ_g 𝑓) ∈ 𝑈})))
53		toponuni 22844	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
54	4, 14, 53	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
55	54	ad2antrr 724	. . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝐵 = ∪ 𝐽)
56	55	ineq1d 4213	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (𝐵 ∩ ∩ ran 𝑔) = (∪ 𝐽 ∩ ∩ ran 𝑔))
57	16	ad2antrr 724	. . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝐽 ∈ Top)
58		simplrl 775	. . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝑔 Fn 𝐴)
59		simplrr 776	. . . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))
60		fvconst2g 7220	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
61	60	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔‘𝑦) ∈ 𝐽))
62	61	ralbidva 3173	. . . . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
64	59, 63	mpbid 231	. . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽)
65		ffnfv 7134	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
66	58, 64, 65	sylanbrc 581	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝑔:𝐴⟶𝐽)
67	66	frnd 6735	. . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ran 𝑔 ⊆ 𝐽)
68	5	ad2antrr 724	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝐴 ∈ Fin)
69		dffn4 6822	. . . . . . . . . . . . . 14 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
70	58, 69	sylib 217	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝑔:𝐴–onto→ran 𝑔)
71		fofi 9372	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑔:𝐴–onto→ran 𝑔) → ran 𝑔 ∈ Fin)
72	68, 70, 71	syl2anc 582	. . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ran 𝑔 ∈ Fin)
73		eqid 2728	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	rintopn 22839	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ran 𝑔 ⊆ 𝐽 ∧ ran 𝑔 ∈ Fin) → (∪ 𝐽 ∩ ∩ ran 𝑔) ∈ 𝐽)
75	57, 67, 72, 74	syl3anc 1368	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (∪ 𝐽 ∩ ∩ ran 𝑔) ∈ 𝐽)
76	56, 75	eqeltrd 2829	. . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (𝐵 ∩ ∩ ran 𝑔) ∈ 𝐽)
77	28	ad2antrr 724	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝑋 ∈ 𝐵)
78		fconstmpt 5744	. . . . . . . . . . . . . 14 ⊢ (𝐴 × {𝑋}) = (𝑦 ∈ 𝐴 ↦ 𝑋)
79		simprl 769	. . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
80	78, 79	eqeltrrid 2834	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
81		mptelixpg 8962	. . . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ((𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
83	80, 82	mpbid 231	. . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦))
84		eleq2 2818	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝑔‘𝑦)))
85	84	ralrn 7103	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → (∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
87	83, 86	mpbird 256	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧)
88		elrint 4998	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧))
89	77, 87, 88	sylanbrc 581	. . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → 𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔))
90	31	inex1 5321	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ ∩ ran 𝑔) ∈ V
91		ixpconstg 8933	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∩ ∩ ran 𝑔) ∈ V) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) = ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴))
92	68, 90, 91	sylancl 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 Σ_g 𝑓) ∈ 𝑈})) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) = ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴))
93		inss2 4232	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
94		fnfvelrn 7095	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ran 𝑔)
95		intss1 4970	. . . . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
98	97	ralrimiva 3143	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → ∀𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
99		ss2ixp 8937	. . . . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
101	92, 100	eqsstrrd 4021	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
102		ssrab 4070	. . . . . . . . . . . . 13 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈} ↔ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ (𝐵 ↑_m 𝐴) ∧ ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σ_g 𝑓) ∈ 𝑈))
103	102	simprbi 495	. . . . . . . . . . . 12 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈} → ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σ_g 𝑓) ∈ 𝑈)
104	103	ad2antll 727	. . . . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σ_g 𝑓) ∈ 𝑈)
105		ssralv 4050	. . . . . . . . . . 11 ⊢ (((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σ_g 𝑓) ∈ 𝑈 → ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴)(𝐺 Σ_g 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 Σ_g 𝑓) ∈ 𝑈})) → ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)
107		eleq2 2818	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑋 ∈ 𝑢 ↔ 𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔)))
108		oveq1 7433	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑢 ↑_m 𝐴) = ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴))
109	108	raleqdv 3323	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))
110	107, 109	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → ((𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈) ↔ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)))
111	110	rspcev 3611	. . . . . . . . . 10 ⊢ (((𝐵 ∩ ∩ ran 𝑔) ∈ 𝐽 ∧ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))
112	76, 89, 106, 111	syl12anc 835	. . . . . . . . 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 Σ_g 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))
113	112	ex 411	. . . . . . . 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 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)))
114	113	3adantr3 1168	. . . . . . 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 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)))
115		eleq2 2818	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
116		sseq1 4007	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈} ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}))
117	115, 116	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈})))
118	117	imbi1d 340	. . . . . . 7 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((((𝐴 × {𝑋}) ∈ 𝑥 ∧ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)) ↔ (((𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ (𝐺 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))))
119	114, 118	syl5ibrcom 246	. . . . . 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 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))))
120	119	expimpd 452	. . . . 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 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))))
121	120	exlimdv 1928	. . . 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 Σ_g 𝑓) ∈ 𝑈}) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))))
122	121	impd 409	. . 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 Σ_g 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)))
123	122	exlimdv 1928	. 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 Σ_g 𝑓) ∈ 𝑈})) → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈)))
124	52, 123	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑_m 𝐴)(𝐺 Σ_g 𝑓) ∈ 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2705 ∀wral 3058 ∃wrex 3067 {crab 3430 Vcvv 3473 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4606 {csn 4632 ∪ cuni 4912 ∩ cint 4953 ↦ cmpt 5235 × cxp 5680 ^◡ccnv 5681 ran crn 5683 “ cima 5685 Fn wfn 6548 ⟶wf 6549 –onto→wfo 6551 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8853 Xcixp 8924 Fincfn 8972 ♯chash 14331 Basecbs 17189 TopOpenctopn 17412 topGenctg 17428 ∏_tcpt 17429 Σ_g cgsu 17431 Mndcmnd 18703 ._gcmg 19037 CMndccmn 19749 Topctop 22823 TopOnctopon 22840 Cn ccn 23156 ↑_ko cxko 23493 TopMndctmd 24002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-rest 17413 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-mre 17575 df-mrc 17576 df-acs 17578 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-mulg 19038 df-cntz 19282 df-cmn 19751 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cn 23159 df-cnp 23160 df-cmp 23319 df-tx 23494 df-xko 23495 df-tmd 24004

This theorem is referenced by: tsmsxp 24087

W3C validator