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Theorem dis2ndc 21984
Description: A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
Assertion
Ref Expression
dis2ndc (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)

Proof of Theorem dis2ndc
Dummy variables 𝑤 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctex 8516 . 2 (𝑋 ≼ ω → 𝑋 ∈ V)
2 pwexr 7479 . 2 (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3 snex 5327 . . . . . . . 8 {𝑥} ∈ V
432a1i 12 . . . . . . 7 (𝑋 ∈ V → (𝑥𝑋 → {𝑥} ∈ V))
5 vex 3502 . . . . . . . . . 10 𝑥 ∈ V
65sneqr 4769 . . . . . . . . 9 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7 sneq 4573 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
86, 7impbii 210 . . . . . . . 8 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
982a1i 12 . . . . . . 7 (𝑋 ∈ V → ((𝑥𝑋𝑦𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
104, 9dom2lem 8541 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1→V)
11 f1f1orn 6622 . . . . . 6 ((𝑥𝑋 ↦ {𝑥}):𝑋1-1→V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥}))
1210, 11syl 17 . . . . 5 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥}))
13 f1oeng 8520 . . . . 5 ((𝑋 ∈ V ∧ (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
1412, 13mpdan 683 . . . 4 (𝑋 ∈ V → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
15 domen1 8651 . . . 4 (𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}) → (𝑋 ≼ ω ↔ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
1614, 15syl 17 . . 3 (𝑋 ∈ V → (𝑋 ≼ ω ↔ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
17 distop 21519 . . . . . . 7 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18 simpr 485 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋) → 𝑥𝑋)
195snelpw 5333 . . . . . . . . . 10 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
2018, 19sylib 219 . . . . . . . . 9 ((𝑋 ∈ V ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
2120fmpttd 6874 . . . . . . . 8 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
2221frnd 6517 . . . . . . 7 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
23 elpwi 4553 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
2423ad2antrl 724 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑦𝑋)
25 simprr 769 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑦)
2624, 25sseldd 3971 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑋)
27 eqidd 2826 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} = {𝑧})
28 sneq 4573 . . . . . . . . . . . 12 (𝑥 = 𝑧 → {𝑥} = {𝑧})
2928rspceeqv 3641 . . . . . . . . . . 11 ((𝑧𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥𝑋 {𝑧} = {𝑥})
3026, 27, 29syl2anc 584 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑥𝑋 {𝑧} = {𝑥})
31 snex 5327 . . . . . . . . . . 11 {𝑧} ∈ V
32 eqid 2825 . . . . . . . . . . . 12 (𝑥𝑋 ↦ {𝑥}) = (𝑥𝑋 ↦ {𝑥})
3332elrnmpt 5826 . . . . . . . . . . 11 ({𝑧} ∈ V → ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥}))
3431, 33ax-mp 5 . . . . . . . . . 10 ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥})
3530, 34sylibr 235 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}))
36 vsnid 4598 . . . . . . . . . 10 𝑧 ∈ {𝑧}
3736a1i 11 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧 ∈ {𝑧})
3825snssd 4740 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ⊆ 𝑦)
39 eleq2 2905 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑧𝑤𝑧 ∈ {𝑧}))
40 sseq1 3995 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑤𝑦 ↔ {𝑧} ⊆ 𝑦))
4139, 40anbi12d 630 . . . . . . . . . 10 (𝑤 = {𝑧} → ((𝑧𝑤𝑤𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
4241rspcev 3626 . . . . . . . . 9 (({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4335, 37, 38, 42syl12anc 834 . . . . . . . 8 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4443ralrimivva 3195 . . . . . . 7 (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
45 basgen2 21513 . . . . . . 7 ((𝒫 𝑋 ∈ Top ∧ ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦)) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
4617, 22, 44, 45syl3anc 1365 . . . . . 6 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
4746adantr 481 . . . . 5 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
4846, 17eqeltrd 2917 . . . . . . 7 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
49 tgclb 21494 . . . . . . 7 (ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
5048, 49sylibr 235 . . . . . 6 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases)
51 2ndci 21972 . . . . . 6 ((ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ 2ndω)
5250, 51sylan 580 . . . . 5 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ 2ndω)
5347, 52eqeltrrd 2918 . . . 4 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
54 is2ndc 21970 . . . . . 6 (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
55 vex 3502 . . . . . . . . 9 𝑏 ∈ V
56 simpr 485 . . . . . . . . . . . . . . 15 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → 𝑥𝑋)
5756, 19sylib 219 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
58 simplrr 774 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
5957, 58eleqtrrd 2920 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ (topGen‘𝑏))
60 vsnid 4598 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
61 tg2 21489 . . . . . . . . . . . . 13 (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦𝑏 (𝑥𝑦𝑦 ⊆ {𝑥}))
6259, 60, 61sylancl 586 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → ∃𝑦𝑏 (𝑥𝑦𝑦 ⊆ {𝑥}))
63 simprrl 777 . . . . . . . . . . . . . . 15 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑥𝑦)
6463snssd 4740 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
65 simprrr 778 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
6664, 65eqssd 3987 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
67 simprl 767 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦𝑏)
6866, 67eqeltrd 2917 . . . . . . . . . . . 12 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
6962, 68rexlimddv 3295 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝑏)
7069fmpttd 6874 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥𝑋 ↦ {𝑥}):𝑋𝑏)
7170frnd 6517 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏)
72 ssdomg 8547 . . . . . . . . 9 (𝑏 ∈ V → (ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏 → ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏))
7355, 71, 72mpsyl 68 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏)
74 simprl 767 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
75 domtr 8554 . . . . . . . 8 ((ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏𝑏 ≼ ω) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
7673, 74, 75syl2anc 584 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
7776rexlimdva2 3291 . . . . . 6 (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
7854, 77syl5bi 243 . . . . 5 (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
7978imp 407 . . . 4 ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
8053, 79impbida 797 . . 3 (𝑋 ∈ V → (ran (𝑥𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
8116, 80bitrd 280 . 2 (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
821, 2, 81pm5.21nii 380 1 (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143  Vcvv 3499  wss 3939  𝒫 cpw 4541  {csn 4563   class class class wbr 5062  cmpt 5142  ran crn 5554  1-1wf1 6348  1-1-ontowf1o 6350  cfv 6351  ωcom 7571  cen 8498  cdom 8499  topGenctg 16703  Topctop 21417  TopBasesctb 21469  2ndωc2ndc 21962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-er 8282  df-en 8502  df-dom 8503  df-topgen 16709  df-top 21418  df-bases 21470  df-2ndc 21964
This theorem is referenced by: (None)
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