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Theorem heiborlem10 36246
Description: Lemma for heibor 36247. The last remaining piece of the proof is to find an element 𝐶 such that 𝐶𝐺0, i.e. 𝐶 is an element of (𝐹‘0) that has no finite subcover, which is true by heiborlem1 36237, since (𝐹‘0) is a finite cover of 𝑋, which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of 𝑈 that covers 𝑋, i.e. 𝑋 is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heibor.4 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6 (𝜑𝐷 ∈ (CMet‘𝑋))
heibor.7 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
Assertion
Ref Expression
heiborlem10 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
Distinct variable groups:   𝑦,𝑛,𝑢,𝐹   𝑚,𝑛,𝑢,𝑣,𝑦,𝑧,𝐷   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑦,𝑧   𝑛,𝐾,𝑦,𝑧   𝜑,𝑣
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)

Proof of Theorem heiborlem10
Dummy variables 𝑡 𝑥 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.7 . . . . . . . 8 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
2 0nn0 12424 . . . . . . . 8 0 ∈ ℕ0
3 inss2 4187 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
4 ffvelcdm 7029 . . . . . . . . 9 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
53, 4sselid 3940 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
61, 2, 5sylancl 586 . . . . . . 7 (𝜑 → (𝐹‘0) ∈ Fin)
7 heibor.8 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
8 fveq2 6839 . . . . . . . . . . . 12 (𝑛 = 0 → (𝐹𝑛) = (𝐹‘0))
9 oveq2 7361 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
108, 9iuneq12d 4980 . . . . . . . . . . 11 (𝑛 = 0 → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1110eqeq2d 2747 . . . . . . . . . 10 (𝑛 = 0 → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
1211rspccva 3578 . . . . . . . . 9 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
137, 2, 12sylancl 586 . . . . . . . 8 (𝜑𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14 eqimss 3998 . . . . . . . 8 (𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) → 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1513, 14syl 17 . . . . . . 7 (𝜑𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
16 heibor.1 . . . . . . . . . 10 𝐽 = (MetOpen‘𝐷)
17 heibor.3 . . . . . . . . . 10 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
18 ovex 7386 . . . . . . . . . 10 (𝑦𝐵0) ∈ V
1916, 17, 18heiborlem1 36237 . . . . . . . . 9 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20 oveq1 7360 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
2120eleq1d 2822 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
2221cbvrexvw 3224 . . . . . . . . 9 (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
2319, 22sylib 217 . . . . . . . 8 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24233expia 1121 . . . . . . 7 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
256, 15, 24syl2anc 584 . . . . . 6 (𝜑 → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
2625adantr 481 . . . . 5 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
27 heibor.4 . . . . . . . . . 10 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
28 vex 3447 . . . . . . . . . 10 𝑥 ∈ V
29 c0ex 11145 . . . . . . . . . 10 0 ∈ V
3016, 17, 27, 28, 29heiborlem2 36238 . . . . . . . . 9 (𝑥𝐺0 ↔ (0 ∈ ℕ0𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾))
31 heibor.5 . . . . . . . . . . . 12 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
32 heibor.6 . . . . . . . . . . . 12 (𝜑𝐷 ∈ (CMet‘𝑋))
3316, 17, 27, 31, 32, 1, 7heiborlem3 36239 . . . . . . . . . . 11 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3433ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3532ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
361ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
377ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
38 simprr 771 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
39 fveq2 6839 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
40 fveq2 6839 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (2nd𝑥) = (2nd𝑡))
4140oveq1d 7368 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((2nd𝑥) + 1) = ((2nd𝑡) + 1))
4239, 41breq12d 5116 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → ((𝑔𝑥)𝐺((2nd𝑥) + 1) ↔ (𝑔𝑡)𝐺((2nd𝑡) + 1)))
43 fveq2 6839 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (𝐵𝑥) = (𝐵𝑡))
4439, 41oveq12d 7371 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝑔𝑥)𝐵((2nd𝑥) + 1)) = ((𝑔𝑡)𝐵((2nd𝑡) + 1)))
4543, 44ineq12d 4171 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) = ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))))
4645eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
4742, 46anbi12d 631 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾)))
4847cbvralvw 3223 . . . . . . . . . . . . . 14 (∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ∀𝑡𝐺 ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
4938, 48sylib 217 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ∀𝑡𝐺 ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
50 simprl 769 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51 eqeq1 2740 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52 oveq1 7360 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
5351, 52ifbieq2d 4510 . . . . . . . . . . . . . . 15 (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
5453cbvmptv 5216 . . . . . . . . . . . . . 14 (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55 seqeq3 13903 . . . . . . . . . . . . . 14 ((𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1))) → seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1)))) = seq0(𝑔, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))))
5654, 55ax-mp 5 . . . . . . . . . . . . 13 seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1)))) = seq0(𝑔, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1))))
57 eqid 2736 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ ⟨(seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))))‘𝑛), (3 / (2↑𝑛))⟩) = (𝑛 ∈ ℕ ↦ ⟨(seq0(𝑔, (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))))‘𝑛), (3 / (2↑𝑛))⟩)
58 simplrl 775 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈𝐽)
59 cmetmet 24634 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60 metxmet 23671 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6116mopnuni 23778 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
6232, 59, 60, 614syl 19 . . . . . . . . . . . . . . . 16 (𝜑𝑋 = 𝐽)
6362adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑋 = 𝐽)
64 simprr 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝐽 = 𝑈)
6563, 64eqtr2d 2777 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈 = 𝑋)
6665adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈 = 𝑋)
6716, 17, 27, 31, 35, 36, 37, 49, 50, 56, 57, 58, 66heiborlem9 36245 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ¬ 𝑋𝐾)
6867expr 457 . . . . . . . . . . 11 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → (∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋𝐾))
6968exlimdv 1936 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → (∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋𝐾))
7034, 69mpd 15 . . . . . . . . 9 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ¬ 𝑋𝐾)
7130, 70sylan2br 595 . . . . . . . 8 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (0 ∈ ℕ0𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋𝐾)
72713exp2 1354 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (0 ∈ ℕ0 → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾))))
732, 72mpi 20 . . . . . 6 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑥 ∈ (𝐹‘0) → ((𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾)))
7473rexlimdv 3148 . . . . 5 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾))
7526, 74syld 47 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 → ¬ 𝑋𝐾))
7675pm2.01d 189 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ¬ 𝑋𝐾)
77 elfvdm 6876 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78 sseq1 3967 . . . . . . . . 9 (𝑢 = 𝑋 → (𝑢 𝑣𝑋 𝑣))
7978rexbidv 3173 . . . . . . . 8 (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8079notbid 317 . . . . . . 7 (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8180, 17elab2g 3630 . . . . . 6 (𝑋 ∈ dom CMet → (𝑋𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8232, 77, 813syl 18 . . . . 5 (𝜑 → (𝑋𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8382adantr 481 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8483con2bid 354 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ¬ 𝑋𝐾))
8576, 84mpbird 256 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣)
8662ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = 𝐽)
8786sseq1d 3973 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 𝑣))
88 inss1 4186 . . . . . . . . 9 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
8988sseli 3938 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
9089elpwid 4567 . . . . . . 7 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣𝑈)
91 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈𝐽)
92 sstr 3950 . . . . . . . 8 ((𝑣𝑈𝑈𝐽) → 𝑣𝐽)
9392unissd 4873 . . . . . . 7 ((𝑣𝑈𝑈𝐽) → 𝑣 𝐽)
9490, 91, 93syl2anr 597 . . . . . 6 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑣 𝐽)
9594biantrud 532 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽)))
96 eqss 3957 . . . . 5 ( 𝐽 = 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽))
9795, 96bitr4di 288 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 𝐽 = 𝑣))
9887, 97bitrd 278 . . 3 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 = 𝑣))
9998rexbidva 3171 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣))
10085, 99mpbid 231 1 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3062  wrex 3071  cin 3907  wss 3908  ifcif 4484  𝒫 cpw 4558  cop 4590   cuni 4863   ciun 4952   class class class wbr 5103  {copab 5165  cmpt 5186  dom cdm 5631  wf 6489  cfv 6493  (class class class)co 7353  cmpo 7355  2nd c2nd 7916  Fincfn 8879  0cc0 11047  1c1 11048   + caddc 11050  cmin 11381   / cdiv 11808  cn 12149  2c2 12204  3c3 12205  0cn0 12409  seqcseq 13898  cexp 13959  ∞Metcxmet 20766  Metcmet 20767  ballcbl 20768  MetOpencmopn 20771  CMetccmet 24602
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 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-inf2 9573  ax-cc 10367  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125
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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-pm 8764  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9374  df-inf 9375  df-oi 9442  df-card 9871  df-acn 9874  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-div 11809  df-nn 12150  df-2 12212  df-3 12213  df-n0 12410  df-z 12496  df-uz 12760  df-q 12866  df-rp 12908  df-xneg 13025  df-xadd 13026  df-xmul 13027  df-ico 13262  df-icc 13263  df-fl 13689  df-seq 13899  df-exp 13960  df-rest 17296  df-topgen 17317  df-psmet 20773  df-xmet 20774  df-met 20775  df-bl 20776  df-mopn 20777  df-fbas 20778  df-fg 20779  df-top 22227  df-topon 22244  df-bases 22280  df-cld 22354  df-ntr 22355  df-cls 22356  df-nei 22433  df-lm 22564  df-haus 22650  df-fil 23181  df-fm 23273  df-flim 23274  df-flf 23275  df-cfil 24603  df-cau 24604  df-cmet 24605
This theorem is referenced by:  heibor  36247
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