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Theorem met1stc 23893
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 23809 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
31mopnuni 23810 . . . . . 6 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
43eleq2d 2820 . . . . 5 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
54biimpar 479 . . . 4 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ βˆͺ 𝐽) β†’ π‘₯ ∈ 𝑋)
6 simpll 766 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
7 simplr 768 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ π‘₯ ∈ 𝑋)
8 nnrp 12931 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ+)
98adantl 483 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ ℝ+)
109rpreccld 12972 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ+)
1110rpxrd 12963 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ*)
121blopn 23872 . . . . . . . . 9 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ*) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ 𝐽)
136, 7, 11, 12syl3anc 1372 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ 𝐽)
1413fmpttd 7064 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):β„•βŸΆπ½)
1514frnd 6677 . . . . . 6 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) βŠ† 𝐽)
16 nnex 12164 . . . . . . . . 9 β„• ∈ V
1716mptex 7174 . . . . . . . 8 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1817rnex 7850 . . . . . . 7 ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1918elpw 4565 . . . . . 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 9587 . . . . . . . . 9 Ο‰ ∈ On
22 nnenom 13891 . . . . . . . . . 10 β„• β‰ˆ Ο‰
2322ensymi 8947 . . . . . . . . 9 Ο‰ β‰ˆ β„•
24 isnumi 9887 . . . . . . . . 9 ((Ο‰ ∈ On ∧ Ο‰ β‰ˆ β„•) β†’ β„• ∈ dom card)
2521, 23, 24mp2an 691 . . . . . . . 8 β„• ∈ dom card
26 ovex 7391 . . . . . . . . . 10 (π‘₯(ballβ€˜π·)(1 / 𝑛)) ∈ V
27 eqid 2733 . . . . . . . . . 10 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) = (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛)))
2826, 27fnmpti 6645 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) Fn β„•
29 dffn4 6763 . . . . . . . . 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 9998 . . . . . . . 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 8956 . . . . . . 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 691 . . . . . 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 766 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
37 simprl 770 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ 𝑧 ∈ 𝐽)
38 simprr 772 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ π‘₯ ∈ 𝑧)
391mopni2 23865 . . . . . . . . . 10 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)
4036, 37, 38, 39syl3anc 1372 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (π‘₯(ballβ€˜π·)π‘Ÿ) βŠ† 𝑧)
41 simp-4l 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†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
42 simp-4r 783 . . . . . . . . . . . 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 770 . . . . . . . . . . . . . 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 12960 . . . . . . . . . . . . 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 12972 . . . . . . . . . . . 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 23782 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ+) β†’ π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
4741, 42, 45, 46syl3anc 1372 . . . . . . . . . . 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 12963 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 12963 . . . . . . . . . . . . 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 12200 . . . . . . . . . . . . . . 15 (𝑦 ∈ β„• β†’ (1 / 𝑦) ∈ ℝ)
5251ad2antrl 727 . . . . . . . . . . . . . 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 12962 . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . 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 11308 . . . . . . . . . . . . 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 23792 . . . . . . . . . . . . 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 1382 . . . . . . . . . . . 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 777 . . . . . . . . . . . 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 3955 . . . . . . . . . . 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 513 . . . . . . . . . 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 12922 . . . . . . . . . . . 12 (π‘Ÿ ∈ ℝ+ ↔ (π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ))
62 nnrecl 12416 . . . . . . . . . . . 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 727 . . . . . . . . . 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 3165 . . . . . . . . 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 3155 . . . . . . . 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 7393 . . . . . . . . 9 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ 𝑦 ∈ β„•) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∈ V)
68 vex 3448 . . . . . . . . . 10 𝑀 ∈ V
69 oveq2 7366 . . . . . . . . . . . . 13 (𝑛 = 𝑦 β†’ (1 / 𝑛) = (1 / 𝑦))
7069oveq2d 7374 . . . . . . . . . . . 12 (𝑛 = 𝑦 β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) = (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7170cbvmptv 5219 . . . . . . . . . . 11 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) = (𝑦 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7271elrnmpt 5912 . . . . . . . . . 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 2823 . . . . . . . . . . 11 (𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦)) β†’ (π‘₯ ∈ 𝑀 ↔ π‘₯ ∈ (π‘₯(ballβ€˜π·)(1 / 𝑦))))
75 sseq1 3970 . . . . . . . . . . 11 (𝑀 = (π‘₯(ballβ€˜π·)(1 / 𝑦)) β†’ (𝑀 βŠ† 𝑧 ↔ (π‘₯(ballβ€˜π·)(1 / 𝑦)) βŠ† 𝑧))
7674, 75anbi12d 632 . . . . . . . . . 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 483 . . . . . . . . 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 5367 . . . . . . . 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 257 . . . . . . 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 458 . . . . . 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 3140 . . . . 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 5109 . . . . . . 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 3309 . . . . . . . . 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 341 . . . . . . . 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 3171 . . . . . . 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 632 . . . . . 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 3580 . . . . 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 836 . . . 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 592 . . 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 3140 . 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 2733 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
9291is1stc2 22809 . 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 584 1 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ 1stΟ‰)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634  ran crn 5635  Oncon0 6318   Fn wfn 6492  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  Ο‰com 7803   β‰ˆ cen 8883   β‰Ό cdom 8884  cardccrd 9876  β„cr 11055  0cc0 11056  1c1 11057  β„*cxr 11193   < clt 11194   ≀ cle 11195   / cdiv 11817  β„•cn 12158  β„+crp 12920  βˆžMetcxmet 20797  ballcbl 20799  MetOpencmopn 20802  Topctop 22258  1stΟ‰c1stc 22804
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-card 9880  df-acn 9883  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-topgen 17330  df-psmet 20804  df-xmet 20805  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-1stc 22806
This theorem is referenced by:  metelcls  24685  metcnp4  24690  metcn4  24691
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