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Theorem heiborlem10 37806
Description: Lemma for heibor 37807. 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 37797, 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 12538 . . . . . . . 8 0 ∈ ℕ0
3 inss2 4245 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
4 ffvelcdm 7100 . . . . . . . . 9 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
53, 4sselid 3992 . . . . . . . 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 6906 . . . . . . . . . . . 12 (𝑛 = 0 → (𝐹𝑛) = (𝐹‘0))
9 oveq2 7438 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
108, 9iuneq12d 5025 . . . . . . . . . . 11 (𝑛 = 0 → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1110eqeq2d 2745 . . . . . . . . . 10 (𝑛 = 0 → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
1211rspccva 3620 . . . . . . . . 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 4053 . . . . . . . 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 7463 . . . . . . . . . 10 (𝑦𝐵0) ∈ V
1916, 17, 18heiborlem1 37797 . . . . . . . . 9 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20 oveq1 7437 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
2120eleq1d 2823 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
2221cbvrexvw 3235 . . . . . . . . 9 (∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾 ↔ ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
2319, 22sylib 218 . . . . . . . 8 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾)
24233expia 1120 . . . . . . 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 480 . . . . 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 3481 . . . . . . . . . 10 𝑥 ∈ V
29 c0ex 11252 . . . . . . . . . 10 0 ∈ V
3016, 17, 27, 28, 29heiborlem2 37798 . . . . . . . . 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 37799 . . . . . . . . . . 11 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3433ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3532ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
361ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
377ad2antrr 726 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
40 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (2nd𝑥) = (2nd𝑡))
4140oveq1d 7445 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((2nd𝑥) + 1) = ((2nd𝑡) + 1))
4239, 41breq12d 5160 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → ((𝑔𝑥)𝐺((2nd𝑥) + 1) ↔ (𝑔𝑡)𝐺((2nd𝑡) + 1)))
43 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (𝐵𝑥) = (𝐵𝑡))
4439, 41oveq12d 7448 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝑔𝑥)𝐵((2nd𝑥) + 1)) = ((𝑔𝑡)𝐵((2nd𝑡) + 1)))
4543, 44ineq12d 4228 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) = ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))))
4645eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → (((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾 ↔ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾))
4742, 46anbi12d 632 . . . . . . . . . . . . . . 15 (𝑥 = 𝑡 → (((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) ↔ ((𝑔𝑡)𝐺((2nd𝑡) + 1) ∧ ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))) ∈ 𝐾)))
4847cbvralvw 3234 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51 eqeq1 2738 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52 oveq1 7437 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
5351, 52ifbieq2d 4556 . . . . . . . . . . . . . . 15 (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
5453cbvmptv 5260 . . . . . . . . . . . . . 14 (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55 seqeq3 14043 . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . 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 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈𝐽)
59 cmetmet 25333 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60 metxmet 24359 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6116mopnuni 24466 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
6232, 59, 60, 614syl 19 . . . . . . . . . . . . . . . 16 (𝜑𝑋 = 𝐽)
6362adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑋 = 𝐽)
64 simprr 773 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝐽 = 𝑈)
6563, 64eqtr2d 2775 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈 = 𝑋)
6665adantr 480 . . . . . . . . . . . . 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 37805 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → ¬ 𝑋𝐾)
6867expr 456 . . . . . . . . . . 11 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → (∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾) → ¬ 𝑋𝐾))
6968exlimdv 1930 . . . . . . . . . 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 1353 . . . . . . 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 3150 . . . . 5 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑥 ∈ (𝐹‘0)(𝑥𝐵0) ∈ 𝐾 → ¬ 𝑋𝐾))
7526, 74syld 47 . . . 4 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (𝑋𝐾 → ¬ 𝑋𝐾))
7675pm2.01d 190 . . 3 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ¬ 𝑋𝐾)
77 elfvdm 6943 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78 sseq1 4020 . . . . . . . . 9 (𝑢 = 𝑋 → (𝑢 𝑣𝑋 𝑣))
7978rexbidv 3176 . . . . . . . 8 (𝑢 = 𝑋 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8079notbid 318 . . . . . . 7 (𝑢 = 𝑋 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣))
8180, 17elab2g 3682 . . . . . 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 480 . . . 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 257 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣)
8662ad2antrr 726 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = 𝐽)
8786sseq1d 4026 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 𝑣))
88 inss1 4244 . . . . . . . . 9 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
8988sseli 3990 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
9089elpwid 4613 . . . . . . 7 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣𝑈)
91 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈𝐽)
92 sstr 4003 . . . . . . . 8 ((𝑣𝑈𝑈𝐽) → 𝑣𝐽)
9392unissd 4921 . . . . . . 7 ((𝑣𝑈𝑈𝐽) → 𝑣 𝐽)
9490, 91, 93syl2anr 597 . . . . . 6 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑣 𝐽)
9594biantrud 531 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽)))
96 eqss 4010 . . . . 5 ( 𝐽 = 𝑣 ↔ ( 𝐽 𝑣 𝑣 𝐽))
9795, 96bitr4di 289 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → ( 𝐽 𝑣 𝐽 = 𝑣))
9887, 97bitrd 279 . . 3 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 = 𝑣))
9998rexbidva 3174 . 2 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑋 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣))
10085, 99mpbid 232 1 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wral 3058  wrex 3067  cin 3961  wss 3962  ifcif 4530  𝒫 cpw 4604  cop 4636   cuni 4911   ciun 4995   class class class wbr 5147  {copab 5209  cmpt 5230  dom cdm 5688  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  2nd c2nd 8011  Fincfn 8983  0cc0 11152  1c1 11153   + caddc 11155  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  3c3 12319  0cn0 12523  seqcseq 14038  cexp 14098  ∞Metcxmet 21366  Metcmet 21367  ballcbl 21368  MetOpencmopn 21371  CMetccmet 25301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ico 13389  df-icc 13390  df-fl 13828  df-seq 14039  df-exp 14099  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-top 22915  df-topon 22932  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lm 23252  df-haus 23338  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-cfil 25302  df-cau 25303  df-cmet 25304
This theorem is referenced by:  heibor  37807
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