Theorem tmdgsum2 21900

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 2622	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↦ (𝐺 Σg 𝑓))
2	1	mptpreima 5628	. . . . . 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 21899	. . . . . . . 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 1326	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
10		tmdgsum2.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
11		cnima 21069	. . . . . . 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 693	. . . . . 6 ⊢ (𝜑 → (◡(𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽 ^ko 𝒫 𝐴))
13	2, 12	syl5eqelr 2706	. . . . 5 ⊢ (𝜑 → {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽 ^ko 𝒫 𝐴))
14	6, 7	tmdtopon 21885	. . . . . . . 8 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15		topontop 20718	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
16	4, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
17		xkopt 21458	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
18	16, 5, 17	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
19		fnconstg 6093	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
20	4, 14, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21		eqid 2622	. . . . . . . 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 21373	. . . . . . 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 693	. . . . . 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 2656	. . . . 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 2703	. . . 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		tmdgsum2.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
27		fconst6g 6094	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (𝐴 × {𝑋}):𝐴⟶𝐵)
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 × {𝑋}):𝐴⟶𝐵)
29		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝐺) ∈ V
30	7, 29	eqeltri 2697	. . . . . . 7 ⊢ 𝐵 ∈ V
31		elmapg 7870	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵 ↑𝑚 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
32	30, 5, 31	sylancr 695	. . . . . 6 ⊢ (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵 ↑𝑚 𝐴) ↔ (𝐴 × {𝑋}):𝐴⟶𝐵))
33	28, 32	mpbird 247	. . . . 5 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵 ↑𝑚 𝐴))
34		fconstmpt 5163	. . . . . . . 8 ⊢ (𝐴 × {𝑋}) = (𝑘 ∈ 𝐴 ↦ 𝑋)
35	34	oveq2i 6661	. . . . . . 7 ⊢ (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))
36		cmnmnd 18208	. . . . . . . . 9 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
37	3, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
38		tmdgsum2.t	. . . . . . . . 9 ⊢ · = (.g‘𝐺)
39	7, 38	gsumconst 18334	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋))
40	37, 5, 26, 39	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋))
41	35, 40	syl5eq 2668	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((#‘𝐴) · 𝑋))
42		tmdgsum2.3	. . . . . 6 ⊢ (𝜑 → ((#‘𝐴) · 𝑋) ∈ 𝑈)
43	41, 42	eqeltrd 2701	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
44		oveq2 6658	. . . . . . 7 ⊢ (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
45	44	eleq1d 2686	. . . . . 6 ⊢ (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
46	45	elrab 3363	. . . . 5 ⊢ ((𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ ((𝐴 × {𝑋}) ∈ (𝐵 ↑𝑚 𝐴) ∧ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
47	33, 43, 46	sylanbrc 698	. . . 4 ⊢ (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
48		tg2 20769	. . . 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}))
49	25, 47, 48	syl2anc 693	. . 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}))
50		eleq2 2690	. . . . 5 ⊢ (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
51		sseq1 3626	. . . . 5 ⊢ (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
52	50, 51	anbi12d 747	. . . 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})))
53	52	rexab2 3373	. . 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})))
54	49, 53	sylib 208	. 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})))
55		toponuni 20719	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
56	4, 14, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
57	56	ad2antrr 762	. . . . . . . . . . . 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)
58	57	ineq1d 3813	. . . . . . . . . . 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))
59	16	ad2antrr 762	. . . . . . . . . . . 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)
60		simplrl 800	. . . . . . . . . . . . . 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)
61		simplrr 801	. . . . . . . . . . . . . . 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))
62		fvconst2g 6467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
63	62	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔‘𝑦) ∈ 𝐽))
64	63	ralbidva 2985	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
65	59, 64	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))
66	61, 65	mpbid 222	. . . . . . . . . . . . . 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)
67		ffnfv 6388	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ 𝐽))
68	60, 66, 67	sylanbrc 698	. . . . . . . . . . . . 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)
69		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐴⟶𝐽 → ran 𝑔 ⊆ 𝐽)
70	68, 69	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})) → ran 𝑔 ⊆ 𝐽)
71	5	ad2antrr 762	. . . . . . . . . . . . 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)
72		dffn4 6121	. . . . . . . . . . . . . 14 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
73	60, 72	sylib 208	. . . . . . . . . . . . 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)
74		fofi 8252	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑔:𝐴–onto→ran 𝑔) → ran 𝑔 ∈ Fin)
75	71, 73, 74	syl2anc 693	. . . . . . . . . . . 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)
76		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
77	76	rintopn 20714	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ran 𝑔 ⊆ 𝐽 ∧ ran 𝑔 ∈ Fin) → (∪ 𝐽 ∩ ∩ ran 𝑔) ∈ 𝐽)
78	59, 70, 75, 77	syl3anc 1326	. . . . . . . . . . 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)
79	58, 78	eqeltrd 2701	. . . . . . . . . 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)
80	26	ad2antrr 762	. . . . . . . . . . 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)
81		fconstmpt 5163	. . . . . . . . . . . . . 14 ⊢ (𝐴 × {𝑋}) = (𝑦 ∈ 𝐴 ↦ 𝑋)
82		simprl 794	. . . . . . . . . . . . . 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))
83	81, 82	syl5eqelr 2706	. . . . . . . . . . . . 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))
84		mptelixpg 7945	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → ((𝑦 ∈ 𝐴 ↦ 𝑋) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
85	71, 84	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)))
86	83, 85	mpbid 222	. . . . . . . . . . . 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))
87		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝑔‘𝑦)))
88	87	ralrn 6362	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑋 ∈ (𝑔‘𝑦)))
89	60, 88	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)))
90	86, 89	mpbird 247	. . . . . . . . . . 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)
91		elrint 4518	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔) ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋 ∈ 𝑧))
92	80, 90, 91	sylanbrc 698	. . . . . . . . . 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))
93	30	inex1 4799	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ ∩ ran 𝑔) ∈ V
94		ixpconstg 7917	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∩ ∩ ran 𝑔) ∈ V) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) = ((𝐵 ∩ ∩ ran 𝑔) ↑𝑚 𝐴))
95	71, 93, 94	sylancl 694	. . . . . . . . . . . 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))
96		inss2 3834	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
97		fnfvelrn 6356	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ran 𝑔)
98		intss1 4492	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘𝑦) ∈ ran 𝑔 → ∩ ran 𝑔 ⊆ (𝑔‘𝑦))
99	97, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ∩ ran 𝑔 ⊆ (𝑔‘𝑦))
100	96, 99	syl5ss 3614	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
101	100	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → ∀𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦))
102		ss2ixp 7921	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑦) → X𝑦 ∈ 𝐴 (𝐵 ∩ ∩ ran 𝑔) ⊆ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
103	60, 101, 102	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))
104	95, 103	eqsstr3d 3640	. . . . . . . . . . 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))
105		ssrab 3680	. . . . . . . . . . . . 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))
106	105	simprbi 480	. . . . . . . . . . . 12 ⊢ (X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} → ∀𝑓 ∈ X 𝑦 ∈ 𝐴 (𝑔‘𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
107	106	ad2antll 765	. . . . . . . . . . 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)
108		ssralv 3666	. . . . . . . . . . 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))
109	104, 107, 108	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)
110		eleq2 2690	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑋 ∈ 𝑢 ↔ 𝑋 ∈ (𝐵 ∩ ∩ ran 𝑔)))
111		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (𝑢 ↑𝑚 𝐴) = ((𝐵 ∩ ∩ ran 𝑔) ↑𝑚 𝐴))
112	111	raleqdv 3144	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝐵 ∩ ∩ ran 𝑔) → (∀𝑓 ∈ (𝑢 ↑𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ∩ ∩ ran 𝑔) ↑𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
113	110, 112	anbi12d 747	. . . . . . . . . . 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)))
114	113	rspcev 3309	. . . . . . . . . 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))
115	79, 92, 109, 114	syl12anc 1324	. . . . . . . . 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))
116	115	ex 450	. . . . . . . 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)))
117	116	3adantr3 1222	. . . . . . 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)))
118		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
119		sseq1 3626	. . . . . . . . 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}))
120	118, 119	anbi12d 747	. . . . . . . 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})))
121	120	imbi1d 331	. . . . . . 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))))
122	117, 121	syl5ibrcom 237	. . . . . 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))))
123	122	expimpd 629	. . . . 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))))
124	123	exlimdv 1861	. . . 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))))
125	124	impd 447	. . 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)))
126	125	exlimdv 1861	. 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)))
127	54, 126	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∩ cint 4475   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  #chash 13117  Basecbs 15857  TopOpenctopn 16082  topGenctg 16098  ∏_tcpt 16099   Σ_g cgsu 16101  Mndcmnd 17294  ._gcmg 17540  CMndccmn 18193  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^_ko cxko 21364  TopMndctmd 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-rest 16083  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-mre 16246  df-mrc 16247  df-acs 16249  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-xko 21366  df-tmd 21876

This theorem is referenced by:  tsmsxp  21958