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Theorem met1stc 24030
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 23946 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
31mopnuni 23947 . . . . . 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 12985 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ+)
98adantl 483 . . . . . . . . . . 11 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ ℝ+)
109rpreccld 13026 . . . . . . . . . 10 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ+)
1110rpxrd 13017 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑛 ∈ β„•) β†’ (1 / 𝑛) ∈ ℝ*)
121blopn 24009 . . . . . . . . 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 7115 . . . . . . 7 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))):β„•βŸΆπ½)
1514frnd 6726 . . . . . 6 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) βŠ† 𝐽)
16 nnex 12218 . . . . . . . . 9 β„• ∈ V
1716mptex 7225 . . . . . . . 8 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1817rnex 7903 . . . . . . 7 ran (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) ∈ V
1918elpw 4607 . . . . . 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 9641 . . . . . . . . 9 Ο‰ ∈ On
22 nnenom 13945 . . . . . . . . . 10 β„• β‰ˆ Ο‰
2322ensymi 9000 . . . . . . . . 9 Ο‰ β‰ˆ β„•
24 isnumi 9941 . . . . . . . . 9 ((Ο‰ ∈ On ∧ Ο‰ β‰ˆ β„•) β†’ β„• ∈ dom card)
2521, 23, 24mp2an 691 . . . . . . . 8 β„• ∈ dom card
26 ovex 7442 . . . . . . . . . 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 6694 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) Fn β„•
29 dffn4 6812 . . . . . . . . 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 10052 . . . . . . . 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 9009 . . . . . . 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 24002 . . . . . . . . . 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 13014 . . . . . . . . . . . . 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 13026 . . . . . . . . . . . 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 23919 . . . . . . . . . . . 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 13017 . . . . . . . . . . . . 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 13017 . . . . . . . . . . . . 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 12254 . . . . . . . . . . . . . . 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 13016 . . . . . . . . . . . . . 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 11362 . . . . . . . . . . . . 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 23929 . . . . . . . . . . . . 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 3993 . . . . . . . . . . 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 12976 . . . . . . . . . . . 12 (π‘Ÿ ∈ ℝ+ ↔ (π‘Ÿ ∈ ℝ ∧ 0 < π‘Ÿ))
62 nnrecl 12470 . . . . . . . . . . . 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 3172 . . . . . . . . 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 3162 . . . . . . . 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 7444 . . . . . . . . 9 ((((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ π‘₯ ∈ 𝑧)) ∧ 𝑦 ∈ β„•) β†’ (π‘₯(ballβ€˜π·)(1 / 𝑦)) ∈ V)
68 vex 3479 . . . . . . . . . 10 𝑀 ∈ V
69 oveq2 7417 . . . . . . . . . . . . 13 (𝑛 = 𝑦 β†’ (1 / 𝑛) = (1 / 𝑦))
7069oveq2d 7425 . . . . . . . . . . . 12 (𝑛 = 𝑦 β†’ (π‘₯(ballβ€˜π·)(1 / 𝑛)) = (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7170cbvmptv 5262 . . . . . . . . . . 11 (𝑛 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑛))) = (𝑦 ∈ β„• ↦ (π‘₯(ballβ€˜π·)(1 / 𝑦)))
7271elrnmpt 5956 . . . . . . . . . 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 4008 . . . . . . . . . . 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 5410 . . . . . . . 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 3147 . . . . 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 5152 . . . . . . 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 3322 . . . . . . . . 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 3178 . . . . . . 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 3613 . . . . 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 3147 . 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 22946 . 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 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  Oncon0 6365   Fn wfn 6539  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   β‰ˆ cen 8936   β‰Ό cdom 8937  cardccrd 9930  β„cr 11109  0cc0 11110  1c1 11111  β„*cxr 11247   < clt 11248   ≀ cle 11249   / cdiv 11871  β„•cn 12212  β„+crp 12974  βˆžMetcxmet 20929  ballcbl 20931  MetOpencmopn 20934  Topctop 22395  1stΟ‰c1stc 22941
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-1stc 22943
This theorem is referenced by:  metelcls  24822  metcnp4  24827  metcn4  24828
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