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Theorem heiborlem10 33944
Description: Lemma for heibor 33945. 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 33935, 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 11508 . . . . . . . 8 0 ∈ ℕ0
3 inss2 3980 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
4 ffvelrn 6500 . . . . . . . . 9 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
53, 4sseldi 3748 . . . . . . . 8 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ Fin)
61, 2, 5sylancl 566 . . . . . . 7 (𝜑 → (𝐹‘0) ∈ Fin)
7 heibor.8 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
8 fveq2 6332 . . . . . . . . . . . 12 (𝑛 = 0 → (𝐹𝑛) = (𝐹‘0))
9 oveq2 6800 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
108, 9iuneq12d 4678 . . . . . . . . . . 11 (𝑛 = 0 → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1110eqeq2d 2780 . . . . . . . . . 10 (𝑛 = 0 → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
1211rspccva 3457 . . . . . . . . 9 ((∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ∧ 0 ∈ ℕ0) → 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
137, 2, 12sylancl 566 . . . . . . . 8 (𝜑𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
14 eqimss 3804 . . . . . . . 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 6822 . . . . . . . . . 10 (𝑦𝐵0) ∈ V
1916, 17, 18heiborlem1 33935 . . . . . . . . 9 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20 oveq1 6799 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
2120eleq1d 2834 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
2221cbvrexv 3320 . . . . . . . . 9 (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
2319, 22sylib 208 . . . . . . . 8 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24233expia 1113 . . . . . . 7 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)) → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
256, 15, 24syl2anc 565 . . . . . 6 (𝜑 → (𝑋𝐾 → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾))
2625adantr 466 . . . . 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 3352 . . . . . . . . . 10 𝑥 ∈ V
29 c0ex 10235 . . . . . . . . . 10 0 ∈ V
3016, 17, 27, 28, 29heiborlem2 33936 . . . . . . . . 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 33937 . . . . . . . . . . 11 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3433ad2antrr 697 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3532ad2antrr 697 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
361ad2antrr 697 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
377ad2antrr 697 . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
40 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (2nd𝑥) = (2nd𝑡))
4140oveq1d 6807 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((2nd𝑥) + 1) = ((2nd𝑡) + 1))
4239, 41breq12d 4797 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → ((𝑔𝑥)𝐺((2nd𝑥) + 1) ↔ (𝑔𝑡)𝐺((2nd𝑡) + 1)))
43 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (𝐵𝑥) = (𝐵𝑡))
4439, 41oveq12d 6810 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝑔𝑥)𝐵((2nd𝑥) + 1)) = ((𝑔𝑡)𝐵((2nd𝑡) + 1)))
4543, 44ineq12d 3964 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) = ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))))
4645eleq1d 2834 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
4742, 46anbi12d 608 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾)))
4847cbvralv 3319 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . 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 746 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51 eqeq1 2774 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52 oveq1 6799 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
5351, 52ifbieq2d 4248 . . . . . . . . . . . . . . 15 (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
5453cbvmptv 4882 . . . . . . . . . . . . . 14 (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55 seqeq3 13012 . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . 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 754 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈𝐽)
59 cmetmet 23302 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60 metxmet 22358 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6116mopnuni 22465 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
6232, 59, 60, 614syl 19 . . . . . . . . . . . . . . . 16 (𝜑𝑋 = 𝐽)
6362adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑋 = 𝐽)
64 simprr 748 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝐽 = 𝑈)
6563, 64eqtr2d 2805 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈 = 𝑋)
6665adantr 466 . . . . . . . . . . . . 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 33943 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ¬ 𝑋𝐾)
6867expr 444 . . . . . . . . . . 11 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → (∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋𝐾))
6968exlimdv 2012 . . . . . . . . . 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 574 . . . . . . . 8 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (0 ∈ ℕ0𝑥 ∈ (𝐹‘0) ∧ (𝑥𝐵0) ∈ 𝐾)) → ¬ 𝑋𝐾)
72713exp2 1446 . . . . . . 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 3177 . . . . 5 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾))
7526, 74syld 47 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 → ¬ 𝑋𝐾))
7675pm2.01d 181 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ¬ 𝑋𝐾)
77 elfvdm 6361 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78 sseq1 3773 . . . . . . . . 9 (𝑢 = 𝑋 → (𝑢 𝑣𝑋 𝑣))
7978rexbidv 3199 . . . . . . . 8 (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8079notbid 307 . . . . . . 7 (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8180, 17elab2g 3502 . . . . . 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 466 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8483con2bid 343 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ¬ 𝑋𝐾))
8576, 84mpbird 247 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣)
8662ad2antrr 697 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = 𝐽)
8786sseq1d 3779 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 𝑣))
88 inss1 3979 . . . . . . . . 9 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
8988sseli 3746 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
9089elpwid 4307 . . . . . . 7 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣𝑈)
91 simprl 746 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈𝐽)
92 sstr 3758 . . . . . . . 8 ((𝑣𝑈𝑈𝐽) → 𝑣𝐽)
9392unissd 4596 . . . . . . 7 ((𝑣𝑈𝑈𝐽) → 𝑣 𝐽)
9490, 91, 93syl2anr 576 . . . . . 6 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑣 𝐽)
9594biantrud 515 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽)))
96 eqss 3765 . . . . 5 ( 𝐽 = 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽))
9795, 96syl6bbr 278 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 𝐽 = 𝑣))
9887, 97bitrd 268 . . 3 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 = 𝑣))
9998rexbidva 3196 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣))
10085, 99mpbid 222 1 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wral 3060  wrex 3061  cin 3720  wss 3721  ifcif 4223  𝒫 cpw 4295  cop 4320   cuni 4572   ciun 4652   class class class wbr 4784  {copab 4844  cmpt 4861  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  2nd c2nd 7313  Fincfn 8108  0cc0 10137  1c1 10138   + caddc 10140  cmin 10467   / cdiv 10885  cn 11221  2c2 11271  3c3 11272  0cn0 11493  seqcseq 13007  cexp 13066  ∞Metcxmt 19945  Metcme 19946  ballcbl 19947  MetOpencmopn 19950  CMetcms 23270
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 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cc 9458  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-acn 8967  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ico 12385  df-icc 12386  df-fl 12800  df-seq 13008  df-exp 13067  df-rest 16290  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-bases 20970  df-cld 21043  df-ntr 21044  df-cls 21045  df-nei 21122  df-lm 21253  df-haus 21339  df-fil 21869  df-fm 21961  df-flim 21962  df-flf 21963  df-cfil 23271  df-cau 23272  df-cmet 23273
This theorem is referenced by:  heibor  33945
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