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Theorem met1stc 24037
Description: The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
methaus.1 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
met1stc (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ 1stΟ‰)

Proof of Theorem met1stc
Dummy variables 𝑛 π‘Ÿ 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 methaus.1 . . 3 𝐽 = (MetOpenβ€˜π·)
21mopntop 23953 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
31mopnuni 23954 . . . . . 6 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
43eleq2d 2819 . . . . 5 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
54biimpar 478 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ 𝑋)
6 simpll 765 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
7 simplr 767 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ π‘₯ ∈ 𝑋)
8 nnrp 12987 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ+)
98adantl 482 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ ℝ+)
109rpreccld 13028 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ+)
1110rpxrd 13019 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ*)
121blopn 24016 . . . . . . . . 9 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ*) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ 𝐽)
136, 7, 11, 12syl3anc 1371 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ 𝐽)
1413fmpttd 7116 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):β„•βŸΆπ½)
1514frnd 6725 . . . . . 6 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) βŠ† 𝐽)
16 nnex 12220 . . . . . . . . 9 β„• ∈ V
1716mptex 7227 . . . . . . . 8 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1817rnex 7905 . . . . . . 7 ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1918elpw 4606 . . . . . 6 (ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ 𝒫 𝐽 ↔ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) βŠ† 𝐽)
2015, 19sylibr 233 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ 𝒫 𝐽)
21 omelon 9643 . . . . . . . . 9 Ο‰ ∈ On
22 nnenom 13947 . . . . . . . . . 10 β„• β‰ˆ Ο‰
2322ensymi 9002 . . . . . . . . 9 Ο‰ β‰ˆ β„•
24 isnumi 9943 . . . . . . . . 9 ((Ο‰ ∈ On ∧ Ο‰ β‰ˆ β„•) β†’ β„• ∈ dom card)
2521, 23, 24mp2an 690 . . . . . . . 8 β„• ∈ dom card
26 ovex 7444 . . . . . . . . . 10 (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ V
27 eqid 2732 . . . . . . . . . 10 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) = (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))
2826, 27fnmpti 6693 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) Fn β„•
29 dffn4 6811 . . . . . . . . 9 ((𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) Fn β„• ↔ (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):ℕ–ontoβ†’ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))))
3028, 29mpbi 229 . . . . . . . 8 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):ℕ–ontoβ†’ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))
31 fodomnum 10054 . . . . . . . 8 (β„• ∈ dom card β†’ ((𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):ℕ–ontoβ†’ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό β„•))
3225, 30, 31mp2 9 . . . . . . 7 ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό β„•
33 domentr 9011 . . . . . . 7 ((ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό β„• ∧ β„• β‰ˆ Ο‰) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰)
3432, 22, 33mp2an 690 . . . . . 6 ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰
3534a1i 11 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰)
36 simpll 765 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
37 simprl 769 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 ∈ 𝐽)
38 simprr 771 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ 𝑧)
391mopni2 24009 . . . . . . . . . 10 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)
4036, 37, 38, 39syl3anc 1371 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)
41 simp-4l 781 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
42 simp-4r 782 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ π‘₯ ∈ 𝑋)
43 simprl 769 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ 𝑦 ∈ β„•)
4443nnrpd 13016 . . . . . . . . . . . . 13 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ 𝑦 ∈ ℝ+)
4544rpreccld 13028 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (1 / 𝑦) ∈ ℝ+)
46 blcntr 23926 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ+) β†’ π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
4741, 42, 45, 46syl3anc 1371 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
4845rpxrd 13019 . . . . . . . . . . . . 13 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (1 / 𝑦) ∈ ℝ*)
49 simplrl 775 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ π‘Ÿ ∈ ℝ+)
5049rpxrd 13019 . . . . . . . . . . . . 13 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ π‘Ÿ ∈ ℝ*)
51 nnrecre 12256 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„• β†’ (1 / 𝑦) ∈ ℝ)
5251ad2antrl 726 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (1 / 𝑦) ∈ ℝ)
5349rpred 13018 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ π‘Ÿ ∈ ℝ)
54 simprr 771 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (1 / 𝑦) < π‘Ÿ)
5552, 53, 54ltled 11364 . . . . . . . . . . . . 13 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (1 / 𝑦) ≀ π‘Ÿ)
56 ssbl 23936 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ ((1 / 𝑦) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ*) ∧ (1 / 𝑦) ≀ π‘Ÿ) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† (π‘₯(ballβ€˜π·)π‘Ÿ))
5741, 42, 48, 50, 55, 56syl221anc 1381 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† (π‘₯(ballβ€˜π·)π‘Ÿ))
58 simplrr 776 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)
5957, 58sstrd 3992 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧)
6047, 59jca 512 . . . . . . . . . 10 (((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) ∧ (𝑦 ∈ β„• ∧ (1 / 𝑦) < π‘Ÿ)) β†’ (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧))
61 elrp 12978 . . . . . . . . . . . 12 (π‘Ÿ ∈ ℝ+ ↔ (π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ))
62 nnrecl 12472 . . . . . . . . . . . 12 ((π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ) β†’ βˆƒπ‘¦ ∈ β„• (1 / 𝑦) < π‘Ÿ)
6361, 62sylbi 216 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ βˆƒπ‘¦ ∈ β„• (1 / 𝑦) < π‘Ÿ)
6463ad2antrl 726 . . . . . . . . . 10 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) β†’ βˆƒπ‘¦ ∈ β„• (1 / 𝑦) < π‘Ÿ)
6560, 64reximddv 3171 . . . . . . . . 9 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ (π‘Ÿ ∈ ℝ+ ∧ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)) β†’ βˆƒπ‘¦ ∈ β„• (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧))
6640, 65rexlimddv 3161 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ βˆƒπ‘¦ ∈ β„• (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧))
67 ovexd 7446 . . . . . . . . 9 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ 𝑦 ∈ β„•) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∈ V)
68 vex 3478 . . . . . . . . . 10 𝑀 ∈ V
69 oveq2 7419 . . . . . . . . . . . . 13 (𝑛 = 𝑦 β†’ (1 / 𝑛) = (1 / 𝑦))
7069oveq2d 7427 . . . . . . . . . . . 12 (𝑛 = 𝑦 β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) = (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7170cbvmptv 5261 . . . . . . . . . . 11 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) = (𝑦 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7271elrnmpt 5955 . . . . . . . . . 10 (𝑀 ∈ V β†’ (𝑀 ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ↔ βˆƒπ‘¦ ∈ β„• 𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦))))
7368, 72mp1i 13 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ (𝑀 ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ↔ βˆƒπ‘¦ ∈ β„• 𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦))))
74 eleq2 2822 . . . . . . . . . . 11 (𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦)) β†’ (π‘₯ ∈ 𝑀 ↔ π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦))))
75 sseq1 4007 . . . . . . . . . . 11 (𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦)) β†’ (𝑀 βŠ† 𝑧 ↔ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧))
7674, 75anbi12d 631 . . . . . . . . . 10 (𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦)) β†’ ((π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧) ↔ (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧)))
7776adantl 482 . . . . . . . . 9 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ 𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦))) β†’ ((π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧) ↔ (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧)))
7867, 73, 77rexxfr2d 5409 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ (βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧) ↔ βˆƒπ‘¦ ∈ β„• (π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∧ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧)))
7966, 78mpbird 256 . . . . . . 7 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))
8079expr 457 . . . . . 6 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑧 ∈ 𝐽) β†’ (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))
8180ralrimiva 3146 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))
82 breq1 5151 . . . . . . 7 (𝑦 = ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ (𝑦 β‰Ό Ο‰ ↔ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰))
83 rexeq 3321 . . . . . . . . 9 (𝑦 = ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ (βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧) ↔ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))
8483imbi2d 340 . . . . . . . 8 (𝑦 = ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ ((π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)) ↔ (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
8584ralbidv 3177 . . . . . . 7 (𝑦 = ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ (βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)) ↔ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
8682, 85anbi12d 631 . . . . . 6 (𝑦 = ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β†’ ((𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))) ↔ (ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))))
8786rspcev 3612 . . . . 5 ((ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ 𝒫 𝐽 ∧ (ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))(π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))) β†’ βˆƒπ‘¦ ∈ 𝒫 𝐽(𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
8820, 35, 81, 87syl12anc 835 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝒫 𝐽(𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
895, 88syldan 591 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ βˆƒπ‘¦ ∈ 𝒫 𝐽(𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
9089ralrimiva 3146 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ βˆͺ π½βˆƒπ‘¦ ∈ 𝒫 𝐽(𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧))))
91 eqid 2732 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
9291is1stc2 22953 . 2 (𝐽 ∈ 1stΟ‰ ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ βˆͺ π½βˆƒπ‘¦ ∈ 𝒫 𝐽(𝑦 β‰Ό Ο‰ ∧ βˆ€π‘§ ∈ 𝐽 (π‘₯ ∈ 𝑧 β†’ βˆƒπ‘€ ∈ 𝑦 (π‘₯ ∈ 𝑀 ∧ 𝑀 βŠ† 𝑧)))))
932, 90, 92sylanbrc 583 1 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ 1stΟ‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677  Oncon0 6364   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7411  Ο‰com 7857   β‰ˆ cen 8938   β‰Ό cdom 8939  cardccrd 9932  β„cr 11111  0cc0 11112  1c1 11113  β„*cxr 11249   < clt 11250   ≀ cle 11251   / cdiv 11873  β„•cn 12214  β„+crp 12976  βˆžMetcxmet 20935  ballcbl 20937  MetOpencmopn 20940  Topctop 22402  1stΟ‰c1stc 22948
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-card 9936  df-acn 9939  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-1stc 22950
This theorem is referenced by:  metelcls  24829  metcnp4  24834  metcn4  24835
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