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Theorem heiborlem10 32585
Description: Lemma for heibor 32586. 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 32576, 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 11154 . . . . . . . 8 0 ∈ ℕ0
3 inss2 3795 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
4 ffvelrn 6250 . . . . . . . . 9 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
53, 4sseldi 3565 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
61, 2, 5sylancl 692 . . . . . . 7 (𝜑 → (𝐹‘0) ∈ Fin)
7 heibor.8 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
8 fveq2 6088 . . . . . . . . . . . 12 (𝑛 = 0 → (𝐹𝑛) = (𝐹‘0))
9 oveq2 6535 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
108, 9iuneq12d 4476 . . . . . . . . . . 11 (𝑛 = 0 → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1110eqeq2d 2619 . . . . . . . . . 10 (𝑛 = 0 → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
1211rspccva 3280 . . . . . . . . 9 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
137, 2, 12sylancl 692 . . . . . . . 8 (𝜑𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14 eqimss 3619 . . . . . . . 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 6555 . . . . . . . . . 10 (𝑦𝐵0) ∈ V
1916, 17, 18heiborlem1 32576 . . . . . . . . 9 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20 oveq1 6534 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
2120eleq1d 2671 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
2221cbvrexv 3147 . . . . . . . . 9 (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
2319, 22sylib 206 . . . . . . . 8 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24233expia 1258 . . . . . . 7 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
256, 15, 24syl2anc 690 . . . . . 6 (𝜑 → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
2625adantr 479 . . . . 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 3175 . . . . . . . . . 10 𝑥 ∈ V
29 c0ex 9890 . . . . . . . . . 10 0 ∈ V
3016, 17, 27, 28, 29heiborlem2 32577 . . . . . . . . 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 32578 . . . . . . . . . . 11 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3433ad2antrr 757 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3532ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
361ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
377ad2antrr 757 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
40 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (2nd𝑥) = (2nd𝑡))
4140oveq1d 6542 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((2nd𝑥) + 1) = ((2nd𝑡) + 1))
4239, 41breq12d 4590 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → ((𝑔𝑥)𝐺((2nd𝑥) + 1) ↔ (𝑔𝑡)𝐺((2nd𝑡) + 1)))
43 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (𝐵𝑥) = (𝐵𝑡))
4439, 41oveq12d 6545 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝑔𝑥)𝐵((2nd𝑥) + 1)) = ((𝑔𝑡)𝐵((2nd𝑡) + 1)))
4543, 44ineq12d 3776 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) = ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))))
4645eleq1d 2671 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
4742, 46anbi12d 742 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾)))
4847cbvralv 3146 . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . 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 789 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51 eqeq1 2613 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
5351, 52ifbieq2d 4060 . . . . . . . . . . . . . . 15 (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
5453cbvmptv 4672 . . . . . . . . . . . . . 14 (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55 seqeq3 12623 . . . . . . . . . . . . . 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 2609 . . . . . . . . . . . . 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 795 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈𝐽)
59 cmetmet 22810 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60 metxmet 21890 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6116mopnuni 21997 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
6232, 59, 60, 614syl 19 . . . . . . . . . . . . . . . 16 (𝜑𝑋 = 𝐽)
6362adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑋 = 𝐽)
64 simprr 791 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝐽 = 𝑈)
6563, 64eqtr2d 2644 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈 = 𝑋)
6665adantr 479 . . . . . . . . . . . . 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 32584 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ¬ 𝑋𝐾)
6867expr 640 . . . . . . . . . . 11 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → (∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋𝐾))
6968exlimdv 1847 . . . . . . . . . 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 491 . . . . . . . 8 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (0 ∈ ℕ0𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋𝐾)
72713exp2 1276 . . . . . . 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 3011 . . . . 5 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾))
7526, 74syld 45 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 → ¬ 𝑋𝐾))
7675pm2.01d 179 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ¬ 𝑋𝐾)
77 elfvdm 6115 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78 sseq1 3588 . . . . . . . . 9 (𝑢 = 𝑋 → (𝑢 𝑣𝑋 𝑣))
7978rexbidv 3033 . . . . . . . 8 (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8079notbid 306 . . . . . . 7 (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8180, 17elab2g 3321 . . . . . 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 479 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8483con2bid 342 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ¬ 𝑋𝐾))
8576, 84mpbird 245 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣)
8662ad2antrr 757 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = 𝐽)
8786sseq1d 3594 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 𝑣))
88 inss1 3794 . . . . . . . . 9 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
8988sseli 3563 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
9089elpwid 4117 . . . . . . 7 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣𝑈)
91 simprl 789 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈𝐽)
92 sstr 3575 . . . . . . . 8 ((𝑣𝑈𝑈𝐽) → 𝑣𝐽)
9392unissd 4392 . . . . . . 7 ((𝑣𝑈𝑈𝐽) → 𝑣 𝐽)
9490, 91, 93syl2anr 493 . . . . . 6 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑣 𝐽)
9594biantrud 526 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽)))
96 eqss 3582 . . . . 5 ( 𝐽 = 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽))
9795, 96syl6bbr 276 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 𝐽 = 𝑣))
9887, 97bitrd 266 . . 3 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 = 𝑣))
9998rexbidva 3030 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣))
10085, 99mpbid 220 1 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wral 2895  wrex 2896  cin 3538  wss 3539  ifcif 4035  𝒫 cpw 4107  cop 4130   cuni 4366   ciun 4449   class class class wbr 4577  {copab 4636  cmpt 4637  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  2nd c2nd 7035  Fincfn 7818  0cc0 9792  1c1 9793   + caddc 9795  cmin 10117   / cdiv 10533  cn 10867  2c2 10917  3c3 10918  0cn0 11139  seqcseq 12618  cexp 12677  ∞Metcxmt 19498  Metcme 19499  ballcbl 19500  MetOpencmopn 19503  CMetcms 22778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cc 9117  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-acn 8628  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ico 12008  df-icc 12009  df-fl 12410  df-seq 12619  df-exp 12678  df-rest 15852  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-top 20463  df-bases 20464  df-topon 20465  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lm 20785  df-haus 20871  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-cfil 22779  df-cau 22780  df-cmet 22781
This theorem is referenced by:  heibor  32586
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