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Theorem tmdgsum2 23447
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 · = (.g𝐺)
tmdgsum2.1 (𝜑𝐺 ∈ CMnd)
tmdgsum2.2 (𝜑𝐺 ∈ TopMnd)
tmdgsum2.a (𝜑𝐴 ∈ Fin)
tmdgsum2.u (𝜑𝑈𝐽)
tmdgsum2.x (𝜑𝑋𝐵)
tmdgsum2.3 (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
Assertion
Ref Expression
tmdgsum2 (𝜑 → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 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.
StepHypRef Expression
1 eqid 2736 . . . . . . 7 (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓))
21mptpreima 6190 . . . . . 6 ((𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) = {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}
3 tmdgsum2.1 . . . . . . . 8 (𝜑𝐺 ∈ CMnd)
4 tmdgsum2.2 . . . . . . . 8 (𝜑𝐺 ∈ TopMnd)
5 tmdgsum2.a . . . . . . . 8 (𝜑𝐴 ∈ Fin)
6 tmdgsum.j . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
7 tmdgsum.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
86, 7tmdgsum 23446 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
93, 4, 5, 8syl3anc 1371 . . . . . . 7 (𝜑 → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
10 tmdgsum2.u . . . . . . 7 (𝜑𝑈𝐽)
11 cnima 22616 . . . . . . 7 (((𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽) ∧ 𝑈𝐽) → ((𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽ko 𝒫 𝐴))
129, 10, 11syl2anc 584 . . . . . 6 (𝜑 → ((𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽ko 𝒫 𝐴))
132, 12eqeltrrid 2843 . . . . 5 (𝜑 → {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽ko 𝒫 𝐴))
146, 7tmdtopon 23432 . . . . . . . 8 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15 topontop 22262 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
164, 14, 153syl 18 . . . . . . 7 (𝜑𝐽 ∈ Top)
17 xkopt 23006 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1816, 5, 17syl2anc 584 . . . . . 6 (𝜑 → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
19 fnconstg 6730 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
204, 14, 193syl 18 . . . . . . 7 (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21 eqid 2736 . . . . . . . 8 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
2221ptval 22921 . . . . . . 7 ((𝐴 ∈ Fin ∧ (𝐴 × {𝐽}) Fn 𝐴) → (∏t‘(𝐴 × {𝐽})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
235, 20, 22syl2anc 584 . . . . . 6 (𝜑 → (∏t‘(𝐴 × {𝐽})) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
2418, 23eqtrd 2776 . . . . 5 (𝜑 → (𝐽ko 𝒫 𝐴) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
2513, 24eleqtrd 2840 . . . 4 (𝜑 → {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
26 oveq2 7365 . . . . . 6 (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
2726eleq1d 2822 . . . . 5 (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
28 tmdgsum2.x . . . . . . 7 (𝜑𝑋𝐵)
29 fconst6g 6731 . . . . . . 7 (𝑋𝐵 → (𝐴 × {𝑋}):𝐴𝐵)
3028, 29syl 17 . . . . . 6 (𝜑 → (𝐴 × {𝑋}):𝐴𝐵)
317fvexi 6856 . . . . . . 7 𝐵 ∈ V
32 elmapg 8778 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵m 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3331, 5, 32sylancr 587 . . . . . 6 (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵m 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3430, 33mpbird 256 . . . . 5 (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵m 𝐴))
35 fconstmpt 5694 . . . . . . . 8 (𝐴 × {𝑋}) = (𝑘𝐴𝑋)
3635oveq2i 7368 . . . . . . 7 (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘𝐴𝑋))
37 cmnmnd 19579 . . . . . . . . 9 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
383, 37syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
39 tmdgsum2.t . . . . . . . . 9 · = (.g𝐺)
407, 39gsumconst 19711 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4138, 5, 28, 40syl3anc 1371 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4236, 41eqtrid 2788 . . . . . 6 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
43 tmdgsum2.3 . . . . . 6 (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
4442, 43eqeltrd 2838 . . . . 5 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
4527, 34, 44elrabd 3647 . . . 4 (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
46 tg2 22315 . . . 4 (({𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}) ∧ (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
4725, 45, 46syl2anc 584 . . 3 (𝜑 → ∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
48 eleq2 2826 . . . . 5 (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
49 sseq1 3969 . . . . 5 (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
5048, 49anbi12d 631 . . . 4 (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
5150rexab2 3657 . . 3 (∃𝑡 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
5247, 51sylib 217 . 2 (𝜑 → ∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
53 toponuni 22263 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
544, 14, 533syl 18 . . . . . . . . . . . . 13 (𝜑𝐵 = 𝐽)
5554ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐵 = 𝐽)
5655ineq1d 4171 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐵 ran 𝑔) = ( 𝐽 ran 𝑔))
5716ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐽 ∈ Top)
58 simplrl 775 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔 Fn 𝐴)
59 simplrr 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))
60 fvconst2g 7151 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
6160eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔𝑦) ∈ 𝐽))
6261ralbidva 3172 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6357, 62syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6459, 63mpbid 231 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽)
65 ffnfv 7066 . . . . . . . . . . . . . 14 (𝑔:𝐴𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6658, 64, 65sylanbrc 583 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔:𝐴𝐽)
6766frnd 6676 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ran 𝑔𝐽)
685ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝐴 ∈ Fin)
69 dffn4 6762 . . . . . . . . . . . . . 14 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
7058, 69sylib 217 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑔:𝐴onto→ran 𝑔)
71 fofi 9282 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
7268, 70, 71syl2anc 584 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ran 𝑔 ∈ Fin)
73 eqid 2736 . . . . . . . . . . . . 13 𝐽 = 𝐽
7473rintopn 22258 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ran 𝑔𝐽 ∧ ran 𝑔 ∈ Fin) → ( 𝐽 ran 𝑔) ∈ 𝐽)
7557, 67, 72, 74syl3anc 1371 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ( 𝐽 ran 𝑔) ∈ 𝐽)
7656, 75eqeltrd 2838 . . . . . . . . . 10 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐵 ran 𝑔) ∈ 𝐽)
7728ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑋𝐵)
78 fconstmpt 5694 . . . . . . . . . . . . . 14 (𝐴 × {𝑋}) = (𝑦𝐴𝑋)
79 simprl 769 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦))
8078, 79eqeltrrid 2843 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (𝑦𝐴𝑋) ∈ X𝑦𝐴 (𝑔𝑦))
81 mptelixpg 8873 . . . . . . . . . . . . . 14 (𝐴 ∈ Fin → ((𝑦𝐴𝑋) ∈ X𝑦𝐴 (𝑔𝑦) ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8268, 81syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ((𝑦𝐴𝑋) ∈ X𝑦𝐴 (𝑔𝑦) ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8380, 82mpbid 231 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦))
84 eleq2 2826 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑦) → (𝑋𝑧𝑋 ∈ (𝑔𝑦)))
8584ralrn 7038 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝑋𝑧 ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8658, 85syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → (∀𝑧 ∈ ran 𝑔 𝑋𝑧 ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8783, 86mpbird 256 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑧 ∈ ran 𝑔 𝑋𝑧)
88 elrint 4952 . . . . . . . . . . 11 (𝑋 ∈ (𝐵 ran 𝑔) ↔ (𝑋𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋𝑧))
8977, 87, 88sylanbrc 583 . . . . . . . . . 10 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑋 ∈ (𝐵 ran 𝑔))
9031inex1 5274 . . . . . . . . . . . . 13 (𝐵 ran 𝑔) ∈ V
91 ixpconstg 8844 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝐵 ran 𝑔) ∈ V) → X𝑦𝐴 (𝐵 ran 𝑔) = ((𝐵 ran 𝑔) ↑m 𝐴))
9268, 90, 91sylancl 586 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → X𝑦𝐴 (𝐵 ran 𝑔) = ((𝐵 ran 𝑔) ↑m 𝐴))
93 inss2 4189 . . . . . . . . . . . . . . 15 (𝐵 ran 𝑔) ⊆ ran 𝑔
94 fnfvelrn 7031 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝐴𝑦𝐴) → (𝑔𝑦) ∈ ran 𝑔)
95 intss1 4924 . . . . . . . . . . . . . . . 16 ((𝑔𝑦) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑦))
9694, 95syl 17 . . . . . . . . . . . . . . 15 ((𝑔 Fn 𝐴𝑦𝐴) → ran 𝑔 ⊆ (𝑔𝑦))
9793, 96sstrid 3955 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝐴𝑦𝐴) → (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
9897ralrimiva 3143 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → ∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
99 ss2ixp 8848 . . . . . . . . . . . . 13 (∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦) → X𝑦𝐴 (𝐵 ran 𝑔) ⊆ X𝑦𝐴 (𝑔𝑦))
10058, 98, 993syl 18 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → X𝑦𝐴 (𝐵 ran 𝑔) ⊆ X𝑦𝐴 (𝑔𝑦))
10192, 100eqsstrrd 3983 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ((𝐵 ran 𝑔) ↑m 𝐴) ⊆ X𝑦𝐴 (𝑔𝑦))
102 ssrab 4030 . . . . . . . . . . . . 13 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ (X𝑦𝐴 (𝑔𝑦) ⊆ (𝐵m 𝐴) ∧ ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈))
103102simprbi 497 . . . . . . . . . . . 12 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} → ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
104103ad2antll 727 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
105 ssralv 4010 . . . . . . . . . . 11 (((𝐵 ran 𝑔) ↑m 𝐴) ⊆ X𝑦𝐴 (𝑔𝑦) → (∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈 → ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
106101, 104, 105sylc 65 . . . . . . . . . 10 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)
107 eleq2 2826 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (𝑋𝑢𝑋 ∈ (𝐵 ran 𝑔)))
108 oveq1 7364 . . . . . . . . . . . . 13 (𝑢 = (𝐵 ran 𝑔) → (𝑢m 𝐴) = ((𝐵 ran 𝑔) ↑m 𝐴))
109108raleqdv 3313 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
110107, 109anbi12d 631 . . . . . . . . . . 11 (𝑢 = (𝐵 ran 𝑔) → ((𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈) ↔ (𝑋 ∈ (𝐵 ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
111110rspcev 3581 . . . . . . . . . 10 (((𝐵 ran 𝑔) ∈ 𝐽 ∧ (𝑋 ∈ (𝐵 ran 𝑔) ∧ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
11276, 89, 106, 111syl12anc 835 . . . . . . . . 9 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
113112ex 413 . . . . . . . 8 ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) → (((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
1141133adantr3 1171 . . . . . . 7 ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦))) → (((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
115 eleq2 2826 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦)))
116 sseq1 3969 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
117115, 116anbi12d 631 . . . . . . . 8 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
118117imbi1d 341 . . . . . . 7 (𝑥 = X𝑦𝐴 (𝑔𝑦) → ((((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)) ↔ (((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
119114, 118syl5ibrcom 246 . . . . . 6 ((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦))) → (𝑥 = X𝑦𝐴 (𝑔𝑦) → (((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
120119expimpd 454 . . . . 5 (𝜑 → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → (((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
121120exlimdv 1936 . . . 4 (𝜑 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → (((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))))
122121impd 411 . . 3 (𝜑 → ((∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
123122exlimdv 1936 . 2 (𝜑 → (∃𝑥(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = ((𝐴 × {𝐽})‘𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ∧ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)))
12452, 123mpd 15 1 (𝜑 → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586   cuni 4865   cint 4907  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  m cmap 8765  Xcixp 8835  Fincfn 8883  chash 14230  Basecbs 17083  TopOpenctopn 17303  topGenctg 17319  tcpt 17320   Σg cgsu 17322  Mndcmnd 18556  .gcmg 18872  CMndccmn 19562  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  ko cxko 22912  TopMndctmd 23421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-rest 17304  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-mre 17466  df-mrc 17467  df-acs 17469  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-xko 22914  df-tmd 23423
This theorem is referenced by:  tsmsxp  23506
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