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Theorem tmdgsum2 22099
 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 2758 . . . . . . 7 (𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓))
21mptpreima 5787 . . . . . 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 22098 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
93, 4, 5, 8syl3anc 1477 . . . . . . 7 (𝜑 → (𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
10 tmdgsum2.u . . . . . . 7 (𝜑𝑈𝐽)
11 cnima 21269 . . . . . . 7 (((𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽) ∧ 𝑈𝐽) → ((𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽 ^ko 𝒫 𝐴))
129, 10, 11syl2anc 696 . . . . . 6 (𝜑 → ((𝑓 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑓)) “ 𝑈) ∈ (𝐽 ^ko 𝒫 𝐴))
132, 12syl5eqelr 2842 . . . . 5 (𝜑 → {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽 ^ko 𝒫 𝐴))
146, 7tmdtopon 22084 . . . . . . . 8 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15 topontop 20918 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
164, 14, 153syl 18 . . . . . . 7 (𝜑𝐽 ∈ Top)
17 xkopt 21658 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1816, 5, 17syl2anc 696 . . . . . 6 (𝜑 → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
19 fnconstg 6252 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
204, 14, 193syl 18 . . . . . . 7 (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21 eqid 2758 . . . . . . . 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 21573 . . . . . . 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 696 . . . . . 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 2792 . . . . 5 (𝜑 → (𝐽 ^ko 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 2839 . . . 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 tmdgsum2.x . . . . . . 7 (𝜑𝑋𝐵)
27 fconst6g 6253 . . . . . . 7 (𝑋𝐵 → (𝐴 × {𝑋}):𝐴𝐵)
2826, 27syl 17 . . . . . 6 (𝜑 → (𝐴 × {𝑋}):𝐴𝐵)
29 fvex 6360 . . . . . . . 8 (Base‘𝐺) ∈ V
307, 29eqeltri 2833 . . . . . . 7 𝐵 ∈ V
31 elmapg 8034 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴 ∈ Fin) → ((𝐴 × {𝑋}) ∈ (𝐵𝑚 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3230, 5, 31sylancr 698 . . . . . 6 (𝜑 → ((𝐴 × {𝑋}) ∈ (𝐵𝑚 𝐴) ↔ (𝐴 × {𝑋}):𝐴𝐵))
3328, 32mpbird 247 . . . . 5 (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵𝑚 𝐴))
34 fconstmpt 5318 . . . . . . . 8 (𝐴 × {𝑋}) = (𝑘𝐴𝑋)
3534oveq2i 6822 . . . . . . 7 (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘𝐴𝑋))
36 cmnmnd 18406 . . . . . . . . 9 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
373, 36syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
38 tmdgsum2.t . . . . . . . . 9 · = (.g𝐺)
397, 38gsumconst 18532 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4037, 5, 26, 39syl3anc 1477 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4135, 40syl5eq 2804 . . . . . 6 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
42 tmdgsum2.3 . . . . . 6 (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
4341, 42eqeltrd 2837 . . . . 5 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
44 oveq2 6819 . . . . . . 7 (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
4544eleq1d 2822 . . . . . 6 (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
4645elrab 3502 . . . . 5 ((𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ ((𝐴 × {𝑋}) ∈ (𝐵𝑚 𝐴) ∧ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
4733, 43, 46sylanbrc 701 . . . 4 (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
48 tg2 20969 . . . 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}))
4925, 47, 48syl2anc 696 . . 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}))
50 eleq2 2826 . . . . 5 (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
51 sseq1 3765 . . . . 5 (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
5250, 51anbi12d 749 . . . 4 (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
5352rexab2 3512 . . 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})))
5449, 53sylib 208 . 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})))
55 toponuni 20919 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
564, 14, 553syl 18 . . . . . . . . . . . . 13 (𝜑𝐵 = 𝐽)
5756ad2antrr 764 . . . . . . . . . . . 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)
5857ineq1d 3954 . . . . . . . . . . 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))
5916ad2antrr 764 . . . . . . . . . . . 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)
60 simplrl 819 . . . . . . . . . . . . . 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)
61 simplrr 820 . . . . . . . . . . . . . . 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))
62 fvconst2g 6629 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
6362eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔𝑦) ∈ 𝐽))
6463ralbidva 3121 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → (∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6559, 64syl 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))
6661, 65mpbid 222 . . . . . . . . . . . . . 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)
67 ffnfv 6549 . . . . . . . . . . . . . 14 (𝑔:𝐴𝐽 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ 𝐽))
6860, 66, 67sylanbrc 701 . . . . . . . . . . . . 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)
69 frn 6212 . . . . . . . . . . . . 13 (𝑔:𝐴𝐽 → ran 𝑔𝐽)
7068, 69syl 17 . . . . . . . . . . . 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)
715ad2antrr 764 . . . . . . . . . . . . 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)
72 dffn4 6280 . . . . . . . . . . . . . 14 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
7360, 72sylib 208 . . . . . . . . . . . . 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)
74 fofi 8415 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
7571, 73, 74syl2anc 696 . . . . . . . . . . . 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)
76 eqid 2758 . . . . . . . . . . . . 13 𝐽 = 𝐽
7776rintopn 20914 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ran 𝑔𝐽 ∧ ran 𝑔 ∈ Fin) → ( 𝐽 ran 𝑔) ∈ 𝐽)
7859, 70, 75, 77syl3anc 1477 . . . . . . . . . . 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)
7958, 78eqeltrd 2837 . . . . . . . . . 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)
8026ad2antrr 764 . . . . . . . . . . 11 (((𝜑 ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦))) ∧ ((𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦) ∧ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})) → 𝑋𝐵)
81 fconstmpt 5318 . . . . . . . . . . . . . 14 (𝐴 × {𝑋}) = (𝑦𝐴𝑋)
82 simprl 811 . . . . . . . . . . . . . 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))
8381, 82syl5eqelr 2842 . . . . . . . . . . . . 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))
84 mptelixpg 8109 . . . . . . . . . . . . . 14 (𝐴 ∈ Fin → ((𝑦𝐴𝑋) ∈ X𝑦𝐴 (𝑔𝑦) ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8571, 84syl 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)))
8683, 85mpbid 222 . . . . . . . . . . . 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))
87 eleq2 2826 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑦) → (𝑋𝑧𝑋 ∈ (𝑔𝑦)))
8887ralrn 6523 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝑋𝑧 ↔ ∀𝑦𝐴 𝑋 ∈ (𝑔𝑦)))
8960, 88syl 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)))
9086, 89mpbird 247 . . . . . . . . . . 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)
91 elrint 4668 . . . . . . . . . . 11 (𝑋 ∈ (𝐵 ran 𝑔) ↔ (𝑋𝐵 ∧ ∀𝑧 ∈ ran 𝑔 𝑋𝑧))
9280, 90, 91sylanbrc 701 . . . . . . . . . 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))
9330inex1 4949 . . . . . . . . . . . . 13 (𝐵 ran 𝑔) ∈ V
94 ixpconstg 8081 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ (𝐵 ran 𝑔) ∈ V) → X𝑦𝐴 (𝐵 ran 𝑔) = ((𝐵 ran 𝑔) ↑𝑚 𝐴))
9571, 93, 94sylancl 697 . . . . . . . . . . . 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))
96 inss2 3975 . . . . . . . . . . . . . . 15 (𝐵 ran 𝑔) ⊆ ran 𝑔
97 fnfvelrn 6517 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝐴𝑦𝐴) → (𝑔𝑦) ∈ ran 𝑔)
98 intss1 4642 . . . . . . . . . . . . . . . 16 ((𝑔𝑦) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑦))
9997, 98syl 17 . . . . . . . . . . . . . . 15 ((𝑔 Fn 𝐴𝑦𝐴) → ran 𝑔 ⊆ (𝑔𝑦))
10096, 99syl5ss 3753 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝐴𝑦𝐴) → (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
101100ralrimiva 3102 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → ∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
102 ss2ixp 8085 . . . . . . . . . . . . 13 (∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦) → X𝑦𝐴 (𝐵 ran 𝑔) ⊆ X𝑦𝐴 (𝑔𝑦))
10360, 101, 1023syl 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))
10495, 103eqsstr3d 3779 . . . . . . . . . . 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))
105 ssrab 3819 . . . . . . . . . . . . 13 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ (X𝑦𝐴 (𝑔𝑦) ⊆ (𝐵𝑚 𝐴) ∧ ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈))
106105simprbi 483 . . . . . . . . . . . 12 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} → ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈)
107106ad2antll 767 . . . . . . . . . . 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)
108 ssralv 3805 . . . . . . . . . . 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))
109104, 107, 108sylc 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)
110 eleq2 2826 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (𝑋𝑢𝑋 ∈ (𝐵 ran 𝑔)))
111 oveq1 6818 . . . . . . . . . . . . 13 (𝑢 = (𝐵 ran 𝑔) → (𝑢𝑚 𝐴) = ((𝐵 ran 𝑔) ↑𝑚 𝐴))
112111raleqdv 3281 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (∀𝑓 ∈ (𝑢𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
113110, 112anbi12d 749 . . . . . . . . . . 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)))
114113rspcev 3447 . . . . . . . . . 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))
11579, 92, 109, 114syl12anc 1475 . . . . . . . . 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))
116115ex 449 . . . . . . . 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)))
1171163adantr3 1177 . . . . . . 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)))
118 eleq2 2826 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦)))
119 sseq1 3765 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
120118, 119anbi12d 749 . . . . . . . 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})))
121120imbi1d 330 . . . . . . 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))))
122117, 121syl5ibrcom 237 . . . . . 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))))
123122expimpd 630 . . . . 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))))
124123exlimdv 2008 . . . 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))))
125124impd 446 . . 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)))
126125exlimdv 2008 . 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)))
12754, 126mpd 15 1 (𝜑 → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1630  ∃wex 1851   ∈ wcel 2137  {cab 2744  ∀wral 3048  ∃wrex 3049  {crab 3052  Vcvv 3338   ∖ cdif 3710   ∩ cin 3712   ⊆ wss 3713  𝒫 cpw 4300  {csn 4319  ∪ cuni 4586  ∩ cint 4625   ↦ cmpt 4879   × cxp 5262  ◡ccnv 5263  ran crn 5265   “ cima 5267   Fn wfn 6042  ⟶wf 6043  –onto→wfo 6045  ‘cfv 6047  (class class class)co 6811   ↑𝑚 cmap 8021  Xcixp 8072  Fincfn 8119  ♯chash 13309  Basecbs 16057  TopOpenctopn 16282  topGenctg 16298  ∏tcpt 16299   Σg cgsu 16301  Mndcmnd 17493  .gcmg 17739  CMndccmn 18391  Topctop 20898  TopOnctopon 20915   Cn ccn 21228   ^ko cxko 21564  TopMndctmd 22073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-fi 8480  df-oi 8578  df-card 8953  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-fzo 12658  df-seq 12994  df-hash 13310  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-rest 16283  df-0g 16302  df-gsum 16303  df-topgen 16304  df-pt 16305  df-mre 16446  df-mrc 16447  df-acs 16449  df-plusf 17440  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-submnd 17535  df-mulg 17740  df-cntz 17948  df-cmn 18393  df-top 20899  df-topon 20916  df-topsp 20937  df-bases 20950  df-cn 21231  df-cnp 21232  df-cmp 21390  df-tx 21565  df-xko 21566  df-tmd 22075 This theorem is referenced by:  tsmsxp  22157
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