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Theorem tmdgsum2 22859
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 2739 . . . . . . 7 (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓))
21mptpreima 6080 . . . . . 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 22858 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
93, 4, 5, 8syl3anc 1372 . . . . . . 7 (𝜑 → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
10 tmdgsum2.u . . . . . . 7 (𝜑𝑈𝐽)
11 cnima 22028 . . . . . . 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 587 . . . . . 6 (𝜑 → ((𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽ko 𝒫 𝐴))
132, 12eqeltrrid 2839 . . . . 5 (𝜑 → {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽ko 𝒫 𝐴))
146, 7tmdtopon 22844 . . . . . . . 8 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15 topontop 21676 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
164, 14, 153syl 18 . . . . . . 7 (𝜑𝐽 ∈ Top)
17 xkopt 22418 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1816, 5, 17syl2anc 587 . . . . . 6 (𝜑 → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
19 fnconstg 6576 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
204, 14, 193syl 18 . . . . . . 7 (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21 eqid 2739 . . . . . . . 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 22333 . . . . . . 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 587 . . . . . 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 2774 . . . . 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 2836 . . . 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 7190 . . . . . 6 (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
2726eleq1d 2818 . . . . 5 (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
28 tmdgsum2.x . . . . . . 7 (𝜑𝑋𝐵)
29 fconst6g 6577 . . . . . . 7 (𝑋𝐵 → (𝐴 × {𝑋}):𝐴𝐵)
3028, 29syl 17 . . . . . 6 (𝜑 → (𝐴 × {𝑋}):𝐴𝐵)
317fvexi 6700 . . . . . . 7 𝐵 ∈ V
32 elmapg 8462 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵m 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3331, 5, 32sylancr 590 . . . . . 6 (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵m 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3430, 33mpbird 260 . . . . 5 (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵m 𝐴))
35 fconstmpt 5595 . . . . . . . 8 (𝐴 × {𝑋}) = (𝑘𝐴𝑋)
3635oveq2i 7193 . . . . . . 7 (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘𝐴𝑋))
37 cmnmnd 19052 . . . . . . . . 9 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
383, 37syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
39 tmdgsum2.t . . . . . . . . 9 · = (.g𝐺)
407, 39gsumconst 19185 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4138, 5, 28, 40syl3anc 1372 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4236, 41syl5eq 2786 . . . . . 6 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
43 tmdgsum2.3 . . . . . 6 (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
4442, 43eqeltrd 2834 . . . . 5 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
4527, 34, 44elrabd 3595 . . . 4 (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
46 tg2 21728 . . . 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 587 . . 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 2822 . . . . 5 (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
49 sseq1 3912 . . . . 5 (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
5048, 49anbi12d 634 . . . 4 (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
5150rexab2 3604 . . 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 221 . 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 21677 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
544, 14, 533syl 18 . . . . . . . . . . . . 13 (𝜑𝐵 = 𝐽)
5554ad2antrr 726 . . . . . . . . . . . 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 4112 . . . . . . . . . . 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 726 . . . . . . . . . . . 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 777 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . 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 6986 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
6160eleq2d 2819 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔𝑦) ∈ 𝐽))
6261ralbidva 3109 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . 14 (𝑔:𝐴𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6658, 64, 65sylanbrc 586 . . . . . . . . . . . . 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 6522 . . . . . . . . . . . 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 726 . . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . 14 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
7058, 69sylib 221 . . . . . . . . . . . . 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 8895 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
7268, 70, 71syl2anc 587 . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 13 𝐽 = 𝐽
7473rintopn 21672 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ran 𝑔𝐽 ∧ ran 𝑔 ∈ Fin) → ( 𝐽 ran 𝑔) ∈ 𝐽)
7557, 67, 72, 74syl3anc 1372 . . . . . . . . . . 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 2834 . . . . . . . . . 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 726 . . . . . . . . . . 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 5595 . . . . . . . . . . . . . 14 (𝐴 × {𝑋}) = (𝑦𝐴𝑋)
79 simprl 771 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . 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 8557 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑦) → (𝑋𝑧𝑋 ∈ (𝑔𝑦)))
8584ralrn 6876 . . . . . . . . . . . . 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 260 . . . . . . . . . . 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 4889 . . . . . . . . . . 11 (𝑋 ∈ (𝐵 ran 𝑔) ↔ (𝑋𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋𝑧))
8977, 87, 88sylanbrc 586 . . . . . . . . . 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 5195 . . . . . . . . . . . . 13 (𝐵 ran 𝑔) ∈ V
91 ixpconstg 8528 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝐵 ran 𝑔) ∈ V) → X𝑦𝐴 (𝐵 ran 𝑔) = ((𝐵 ran 𝑔) ↑m 𝐴))
9268, 90, 91sylancl 589 . . . . . . . . . . . 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 4130 . . . . . . . . . . . . . . 15 (𝐵 ran 𝑔) ⊆ ran 𝑔
94 fnfvelrn 6870 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝐴𝑦𝐴) → (𝑔𝑦) ∈ ran 𝑔)
95 intss1 4861 . . . . . . . . . . . . . . . 16 ((𝑔𝑦) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑦))
9694, 95syl 17 . . . . . . . . . . . . . . 15 ((𝑔 Fn 𝐴𝑦𝐴) → ran 𝑔 ⊆ (𝑔𝑦))
9793, 96sstrid 3898 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝐴𝑦𝐴) → (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
9897ralrimiva 3097 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → ∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
99 ss2ixp 8532 . . . . . . . . . . . . 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 3926 . . . . . . . . . . 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 3972 . . . . . . . . . . . . 13 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ (X𝑦𝐴 (𝑔𝑦) ⊆ (𝐵m 𝐴) ∧ ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈))
103102simprbi 500 . . . . . . . . . . . 12 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} → ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
104103ad2antll 729 . . . . . . . . . . 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 3953 . . . . . . . . . . 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 2822 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (𝑋𝑢𝑋 ∈ (𝐵 ran 𝑔)))
108 oveq1 7189 . . . . . . . . . . . . 13 (𝑢 = (𝐵 ran 𝑔) → (𝑢m 𝐴) = ((𝐵 ran 𝑔) ↑m 𝐴))
109108raleqdv 3317 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
110107, 109anbi12d 634 . . . . . . . . . . 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 3529 . . . . . . . . . 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 836 . . . . . . . . 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 416 . . . . . . . 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 1172 . . . . . . 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 2822 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦)))
116 sseq1 3912 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
117115, 116anbi12d 634 . . . . . . . 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 345 . . . . . . 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 250 . . . . . 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 457 . . . . 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 1940 . . . 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 414 . . 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 1940 . 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 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  {cab 2717  wral 3054  wrex 3055  {crab 3058  Vcvv 3400  cdif 3850  cin 3852  wss 3853  𝒫 cpw 4498  {csn 4526   cuni 4806   cint 4846  cmpt 5120   × cxp 5533  ccnv 5534  ran crn 5536  cima 5538   Fn wfn 6344  wf 6345  ontowfo 6347  cfv 6349  (class class class)co 7182  m cmap 8449  Xcixp 8519  Fincfn 8567  chash 13794  Basecbs 16598  TopOpenctopn 16810  topGenctg 16826  tcpt 16827   Σg cgsu 16829  Mndcmnd 18039  .gcmg 18354  CMndccmn 19036  Topctop 21656  TopOnctopon 21673   Cn ccn 21987  ko cxko 22324  TopMndctmd 22833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-iin 4894  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-se 5494  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-of 7437  df-om 7612  df-1st 7726  df-2nd 7727  df-supp 7869  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-er 8332  df-map 8451  df-ixp 8520  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-fsupp 8919  df-fi 8960  df-oi 9059  df-card 9453  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-n0 11989  df-z 12075  df-uz 12337  df-fz 12994  df-fzo 13137  df-seq 13473  df-hash 13795  df-ndx 16601  df-slot 16602  df-base 16604  df-sets 16605  df-ress 16606  df-plusg 16693  df-rest 16811  df-0g 16830  df-gsum 16831  df-topgen 16832  df-pt 16833  df-mre 16972  df-mrc 16973  df-acs 16975  df-plusf 17979  df-mgm 17980  df-sgrp 18029  df-mnd 18040  df-submnd 18085  df-mulg 18355  df-cntz 18577  df-cmn 19038  df-top 21657  df-topon 21674  df-topsp 21696  df-bases 21709  df-cn 21990  df-cnp 21991  df-cmp 22150  df-tx 22325  df-xko 22326  df-tmd 22835
This theorem is referenced by:  tsmsxp  22918
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