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Theorem heiborlem10 35987
Description: Lemma for heibor 35988. 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 35978, 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 12257 . . . . . . . 8 0 ∈ ℕ0
3 inss2 4164 . . . . . . . . 9 (𝒫 𝑋 ∩ Fin) ⊆ Fin
4 ffvelrn 6968 . . . . . . . . 9 ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ (𝒫 𝑋 ∩ Fin))
53, 4sselid 3920 . . . . . . . 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 6783 . . . . . . . . . . . 12 (𝑛 = 0 → (𝐹𝑛) = (𝐹‘0))
9 oveq2 7292 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑦𝐵𝑛) = (𝑦𝐵0))
108, 9iuneq12d 4953 . . . . . . . . . . 11 (𝑛 = 0 → 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0))
1110eqeq2d 2750 . . . . . . . . . 10 (𝑛 = 0 → (𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛) ↔ 𝑋 = 𝑦 ∈ (𝐹‘0)(𝑦𝐵0)))
1211rspccva 3561 . . . . . . . . 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 3978 . . . . . . . 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 7317 . . . . . . . . . 10 (𝑦𝐵0) ∈ V
1916, 17, 18heiborlem1 35978 . . . . . . . . 9 (((𝐹‘0) ∈ Fin ∧ 𝑋 𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∧ 𝑋𝐾) → ∃𝑦 ∈ (𝐹‘0)(𝑦𝐵0) ∈ 𝐾)
20 oveq1 7291 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐵0) = (𝑥𝐵0))
2120eleq1d 2824 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐵0) ∈ 𝐾 ↔ (𝑥𝐵0) ∈ 𝐾))
2221cbvrexvw 3385 . . . . . . . . 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 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 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 3437 . . . . . . . . . 10 𝑥 ∈ V
29 c0ex 10978 . . . . . . . . . 10 0 ∈ V
3016, 17, 27, 28, 29heiborlem2 35979 . . . . . . . . 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 35980 . . . . . . . . . . 11 (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3433ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑥𝐺0) → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
3532ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐷 ∈ (CMet‘𝑋))
361ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
377ad2antrr 723 . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . 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 6783 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
40 fveq2 6783 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (2nd𝑥) = (2nd𝑡))
4140oveq1d 7299 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((2nd𝑥) + 1) = ((2nd𝑡) + 1))
4239, 41breq12d 5088 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑡 → ((𝑔𝑥)𝐺((2nd𝑥) + 1) ↔ (𝑔𝑡)𝐺((2nd𝑡) + 1)))
43 fveq2 6783 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → (𝐵𝑥) = (𝐵𝑡))
4439, 41oveq12d 7302 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑡 → ((𝑔𝑥)𝐵((2nd𝑥) + 1)) = ((𝑔𝑡)𝐵((2nd𝑡) + 1)))
4543, 44ineq12d 4148 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑡 → ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) = ((𝐵𝑡) ∩ ((𝑔𝑡)𝐵((2nd𝑡) + 1))))
4645eleq1d 2824 . . . . . . . . . . . . . . . 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 3384 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑥𝐺0)
51 eqeq1 2743 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 = 0 ↔ 𝑚 = 0))
52 oveq1 7291 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑚 → (𝑔 − 1) = (𝑚 − 1))
5351, 52ifbieq2d 4486 . . . . . . . . . . . . . . 15 (𝑔 = 𝑚 → if(𝑔 = 0, 𝑥, (𝑔 − 1)) = if(𝑚 = 0, 𝑥, (𝑚 − 1)))
5453cbvmptv 5188 . . . . . . . . . . . . . 14 (𝑔 ∈ ℕ0 ↦ if(𝑔 = 0, 𝑥, (𝑔 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝑥, (𝑚 − 1)))
55 seqeq3 13735 . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ (𝑥𝐺0 ∧ ∀𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))) → 𝑈𝐽)
59 cmetmet 24459 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
60 metxmet 23496 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6116mopnuni 23603 . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝐽 = 𝑈)
6563, 64eqtr2d 2780 . . . . . . . . . . . . . 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 35986 . . . . . . . . . . . 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 1937 . . . . . . . . . 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 3213 . . . . 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 6815 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
78 sseq1 3947 . . . . . . . . 9 (𝑢 = 𝑋 → (𝑢 𝑣𝑋 𝑣))
7978rexbidv 3227 . . . . . . . 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 3612 . . . . . 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 355 . . 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 723 . . . . 5 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → 𝑋 = 𝐽)
8786sseq1d 3953 . . . 4 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 𝑣))
88 inss1 4163 . . . . . . . . 9 (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
8988sseli 3918 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣 ∈ 𝒫 𝑈)
9089elpwid 4545 . . . . . . 7 (𝑣 ∈ (𝒫 𝑈 ∩ Fin) → 𝑣𝑈)
91 simprl 768 . . . . . . 7 ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → 𝑈𝐽)
92 sstr 3930 . . . . . . . 8 ((𝑣𝑈𝑈𝐽) → 𝑣𝐽)
9392unissd 4850 . . . . . . 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 3937 . . . . 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 278 . . 3 (((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) ∧ 𝑣 ∈ (𝒫 𝑈 ∩ Fin)) → (𝑋 𝑣 𝐽 = 𝑣))
9998rexbidva 3226 . 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 1086   = wceq 1539  wex 1782  wcel 2107  {cab 2716  wral 3065  wrex 3066  cin 3887  wss 3888  ifcif 4460  𝒫 cpw 4534  cop 4568   cuni 4840   ciun 4925   class class class wbr 5075  {copab 5137  cmpt 5158  dom cdm 5590  wf 6433  cfv 6437  (class class class)co 7284  cmpo 7286  2nd c2nd 7839  Fincfn 8742  0cc0 10880  1c1 10881   + caddc 10883  cmin 11214   / cdiv 11641  cn 11982  2c2 12037  3c3 12038  0cn0 12242  seqcseq 13730  cexp 13791  ∞Metcxmet 20591  Metcmet 20592  ballcbl 20593  MetOpencmopn 20596  CMetccmet 24427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-inf2 9408  ax-cc 10200  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-map 8626  df-pm 8627  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-inf 9211  df-oi 9278  df-card 9706  df-acn 9709  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-n0 12243  df-z 12329  df-uz 12592  df-q 12698  df-rp 12740  df-xneg 12857  df-xadd 12858  df-xmul 12859  df-ico 13094  df-icc 13095  df-fl 13521  df-seq 13731  df-exp 13792  df-rest 17142  df-topgen 17163  df-psmet 20598  df-xmet 20599  df-met 20600  df-bl 20601  df-mopn 20602  df-fbas 20603  df-fg 20604  df-top 22052  df-topon 22069  df-bases 22105  df-cld 22179  df-ntr 22180  df-cls 22181  df-nei 22258  df-lm 22389  df-haus 22475  df-fil 23006  df-fm 23098  df-flim 23099  df-flf 23100  df-cfil 24428  df-cau 24429  df-cmet 24430
This theorem is referenced by:  heibor  35988
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