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Theorem tmdgsum2 24120
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 2735 . . . . . . 7 (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓))
21mptpreima 6260 . . . . . 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 24119 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
93, 4, 5, 8syl3anc 1370 . . . . . . 7 (𝜑 → (𝑓 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑓)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
10 tmdgsum2.u . . . . . . 7 (𝜑𝑈𝐽)
11 cnima 23289 . . . . . . 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 2844 . . . . 5 (𝜑 → {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ∈ (𝐽ko 𝒫 𝐴))
146, 7tmdtopon 24105 . . . . . . . 8 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
15 topontop 22935 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
164, 14, 153syl 18 . . . . . . 7 (𝜑𝐽 ∈ Top)
17 xkopt 23679 . . . . . . 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 6797 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝐵) → (𝐴 × {𝐽}) Fn 𝐴)
204, 14, 193syl 18 . . . . . . 7 (𝜑 → (𝐴 × {𝐽}) Fn 𝐴)
21 eqid 2735 . . . . . . . 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 23594 . . . . . . 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 2775 . . . . 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 2841 . . . 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 7439 . . . . . 6 (𝑓 = (𝐴 × {𝑋}) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐴 × {𝑋})))
2726eleq1d 2824 . . . . 5 (𝑓 = (𝐴 × {𝑋}) → ((𝐺 Σg 𝑓) ∈ 𝑈 ↔ (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈))
28 tmdgsum2.x . . . . . . 7 (𝜑𝑋𝐵)
29 fconst6g 6798 . . . . . . 7 (𝑋𝐵 → (𝐴 × {𝑋}):𝐴𝐵)
3028, 29syl 17 . . . . . 6 (𝜑 → (𝐴 × {𝑋}):𝐴𝐵)
317fvexi 6921 . . . . . . 7 𝐵 ∈ V
32 elmapg 8878 . . . . . . 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 257 . . . . 5 (𝜑 → (𝐴 × {𝑋}) ∈ (𝐵m 𝐴))
35 fconstmpt 5751 . . . . . . . 8 (𝐴 × {𝑋}) = (𝑘𝐴𝑋)
3635oveq2i 7442 . . . . . . 7 (𝐺 Σg (𝐴 × {𝑋})) = (𝐺 Σg (𝑘𝐴𝑋))
37 cmnmnd 19830 . . . . . . . . 9 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
383, 37syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
39 tmdgsum2.t . . . . . . . . 9 · = (.g𝐺)
407, 39gsumconst 19967 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4138, 5, 28, 40syl3anc 1370 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
4236, 41eqtrid 2787 . . . . . 6 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) = ((♯‘𝐴) · 𝑋))
43 tmdgsum2.3 . . . . . 6 (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)
4442, 43eqeltrd 2839 . . . . 5 (𝜑 → (𝐺 Σg (𝐴 × {𝑋})) ∈ 𝑈)
4527, 34, 44elrabd 3697 . . . 4 (𝜑 → (𝐴 × {𝑋}) ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})
46 tg2 22988 . . . 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 2828 . . . . 5 (𝑡 = 𝑥 → ((𝐴 × {𝑋}) ∈ 𝑡 ↔ (𝐴 × {𝑋}) ∈ 𝑥))
49 sseq1 4021 . . . . 5 (𝑡 = 𝑥 → (𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ 𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
5048, 49anbi12d 632 . . . 4 (𝑡 = 𝑥 → (((𝐴 × {𝑋}) ∈ 𝑡𝑡 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}) ↔ ((𝐴 × {𝑋}) ∈ 𝑥𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈})))
5150rexab2 3708 . . 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 218 . 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 22936 . . . . . . . . . . . . . 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 4227 . . . . . . . . . . 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 7222 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝐴 × {𝐽})‘𝑦) = 𝐽)
6160eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑦𝐴) → ((𝑔𝑦) ∈ ((𝐴 × {𝐽})‘𝑦) ↔ (𝑔𝑦) ∈ 𝐽))
6261ralbidva 3174 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 7139 . . . . . . . . . . . . . 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 6745 . . . . . . . . . . . 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 6827 . . . . . . . . . . . . . 14 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
7058, 69sylib 218 . . . . . . . . . . . . 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 9349 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 𝐽 = 𝐽
7473rintopn 22931 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ran 𝑔𝐽 ∧ ran 𝑔 ∈ Fin) → ( 𝐽 ran 𝑔) ∈ 𝐽)
7557, 67, 72, 74syl3anc 1370 . . . . . . . . . . 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 2839 . . . . . . . . . 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 5751 . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . 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 8974 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑦) → (𝑋𝑧𝑋 ∈ (𝑔𝑦)))
8584ralrn 7108 . . . . . . . . . . . . 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 257 . . . . . . . . . . 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 4994 . . . . . . . . . . 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 5323 . . . . . . . . . . . . 13 (𝐵 ran 𝑔) ∈ V
91 ixpconstg 8945 . . . . . . . . . . . . 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 4246 . . . . . . . . . . . . . . 15 (𝐵 ran 𝑔) ⊆ ran 𝑔
94 fnfvelrn 7100 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝐴𝑦𝐴) → (𝑔𝑦) ∈ ran 𝑔)
95 intss1 4968 . . . . . . . . . . . . . . . 16 ((𝑔𝑦) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑦))
9694, 95syl 17 . . . . . . . . . . . . . . 15 ((𝑔 Fn 𝐴𝑦𝐴) → ran 𝑔 ⊆ (𝑔𝑦))
9793, 96sstrid 4007 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝐴𝑦𝐴) → (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
9897ralrimiva 3144 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → ∀𝑦𝐴 (𝐵 ran 𝑔) ⊆ (𝑔𝑦))
99 ss2ixp 8949 . . . . . . . . . . . . 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 4035 . . . . . . . . . . 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 4083 . . . . . . . . . . . . 13 (X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ (X𝑦𝐴 (𝑔𝑦) ⊆ (𝐵m 𝐴) ∧ ∀𝑓X 𝑦𝐴 (𝑔𝑦)(𝐺 Σg 𝑓) ∈ 𝑈))
103102simprbi 496 . . . . . . . . . . . 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 4064 . . . . . . . . . . 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 2828 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (𝑋𝑢𝑋 ∈ (𝐵 ran 𝑔)))
108 oveq1 7438 . . . . . . . . . . . . 13 (𝑢 = (𝐵 ran 𝑔) → (𝑢m 𝐴) = ((𝐵 ran 𝑔) ↑m 𝐴))
109108raleqdv 3324 . . . . . . . . . . . 12 (𝑢 = (𝐵 ran 𝑔) → (∀𝑓 ∈ (𝑢m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈 ↔ ∀𝑓 ∈ ((𝐵 ran 𝑔) ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
110107, 109anbi12d 632 . . . . . . . . . . 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 3622 . . . . . . . . . 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 837 . . . . . . . . 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 412 . . . . . . . 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 1170 . . . . . . 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 2828 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → ((𝐴 × {𝑋}) ∈ 𝑥 ↔ (𝐴 × {𝑋}) ∈ X𝑦𝐴 (𝑔𝑦)))
116 sseq1 4021 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈} ↔ X𝑦𝐴 (𝑔𝑦) ⊆ {𝑓 ∈ (𝐵m 𝐴) ∣ (𝐺 Σg 𝑓) ∈ 𝑈}))
117115, 116anbi12d 632 . . . . . . . 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 247 . . . . . 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 453 . . . . 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 1931 . . . 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 410 . . 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 1931 . 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912   cint 4951  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  m cmap 8865  Xcixp 8936  Fincfn 8984  chash 14366  Basecbs 17245  TopOpenctopn 17468  topGenctg 17484  tcpt 17485   Σg cgsu 17487  Mndcmnd 18760  .gcmg 19098  CMndccmn 19813  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  ko cxko 23585  TopMndctmd 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-rest 17469  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-mre 17631  df-mrc 17632  df-acs 17634  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-xko 23587  df-tmd 24096
This theorem is referenced by:  tsmsxp  24179
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