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Theorem dis2ndc 22611
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 8753 . 2 (𝑋 ≼ ω → 𝑋 ∈ V)
2 pwexr 7615 . 2 (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3 snex 5354 . . . . . . . 8 {𝑥} ∈ V
432a1i 12 . . . . . . 7 (𝑋 ∈ V → (𝑥𝑋 → {𝑥} ∈ V))
5 vex 3436 . . . . . . . . . 10 𝑥 ∈ V
65sneqr 4771 . . . . . . . . 9 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7 sneq 4571 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
86, 7impbii 208 . . . . . . . 8 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
982a1i 12 . . . . . . 7 (𝑋 ∈ V → ((𝑥𝑋𝑦𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
104, 9dom2lem 8780 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1→V)
11 f1f1orn 6727 . . . . . 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 8759 . . . . 5 ((𝑋 ∈ V ∧ (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
1412, 13mpdan 684 . . . 4 (𝑋 ∈ V → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
15 domen1 8906 . . . 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 22145 . . . . . . 7 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18 simpr 485 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋) → 𝑥𝑋)
195snelpw 5361 . . . . . . . . . 10 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
2018, 19sylib 217 . . . . . . . . 9 ((𝑋 ∈ V ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
2120fmpttd 6989 . . . . . . . 8 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
2221frnd 6608 . . . . . . 7 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
23 elpwi 4542 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
2423ad2antrl 725 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑦𝑋)
25 simprr 770 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑦)
2624, 25sseldd 3922 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑋)
27 eqidd 2739 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} = {𝑧})
28 sneq 4571 . . . . . . . . . . . 12 (𝑥 = 𝑧 → {𝑥} = {𝑧})
2928rspceeqv 3575 . . . . . . . . . . 11 ((𝑧𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥𝑋 {𝑧} = {𝑥})
3026, 27, 29syl2anc 584 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑥𝑋 {𝑧} = {𝑥})
31 snex 5354 . . . . . . . . . . 11 {𝑧} ∈ V
32 eqid 2738 . . . . . . . . . . . 12 (𝑥𝑋 ↦ {𝑥}) = (𝑥𝑋 ↦ {𝑥})
3332elrnmpt 5865 . . . . . . . . . . 11 ({𝑧} ∈ V → ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥}))
3431, 33ax-mp 5 . . . . . . . . . 10 ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥})
3530, 34sylibr 233 . . . . . . . . 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 4742 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ⊆ 𝑦)
39 eleq2 2827 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑧𝑤𝑧 ∈ {𝑧}))
40 sseq1 3946 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑤𝑦 ↔ {𝑧} ⊆ 𝑦))
4139, 40anbi12d 631 . . . . . . . . . 10 (𝑤 = {𝑧} → ((𝑧𝑤𝑤𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
4241rspcev 3561 . . . . . . . . 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 3123 . . . . . . 7 (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
45 basgen2 22139 . . . . . . 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 1370 . . . . . 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 2839 . . . . . . 7 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
49 tgclb 22120 . . . . . . 7 (ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
5048, 49sylibr 233 . . . . . 6 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases)
51 2ndci 22599 . . . . . 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 2840 . . . 4 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
54 is2ndc 22597 . . . . . 6 (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
55 vex 3436 . . . . . . . . 9 𝑏 ∈ V
56 simpr 485 . . . . . . . . . . . . . . 15 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → 𝑥𝑋)
5756, 19sylib 217 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
58 simplrr 775 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
5957, 58eleqtrrd 2842 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ (topGen‘𝑏))
60 vsnid 4598 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
61 tg2 22115 . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . 15 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑥𝑦)
6463snssd 4742 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
65 simprrr 779 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
6664, 65eqssd 3938 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
67 simprl 768 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦𝑏)
6866, 67eqeltrd 2839 . . . . . . . . . . . 12 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
6962, 68rexlimddv 3220 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝑏)
7069fmpttd 6989 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥𝑋 ↦ {𝑥}):𝑋𝑏)
7170frnd 6608 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏)
72 ssdomg 8786 . . . . . . . . 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 768 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
75 domtr 8793 . . . . . . . 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 3216 . . . . . 6 (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
7854, 77syl5bi 241 . . . . 5 (𝑋 ∈ V → (𝒫 𝑋 ∈ 2ndω → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
7978imp 407 . . . 4 ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndω) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
8053, 79impbida 798 . . 3 (𝑋 ∈ V → (ran (𝑥𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω))
8116, 80bitrd 278 . 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  wss 3887  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  cmpt 5157  ran crn 5590  1-1wf1 6430  1-1-ontowf1o 6432  cfv 6433  ωcom 7712  cen 8730  cdom 8731  topGenctg 17148  Topctop 22042  TopBasesctb 22095  2ndωc2ndc 22589
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-er 8498  df-en 8734  df-dom 8735  df-topgen 17154  df-top 22043  df-bases 22096  df-2ndc 22591
This theorem is referenced by: (None)
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