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Theorem dis2ndc 22811
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 8903 . 2 (𝑋 ≼ ω → 𝑋 ∈ V)
2 pwexr 7699 . 2 (𝒫 𝑋 ∈ 2ndω → 𝑋 ∈ V)
3 vsnex 5386 . . . . . . . 8 {𝑥} ∈ V
432a1i 12 . . . . . . 7 (𝑋 ∈ V → (𝑥𝑋 → {𝑥} ∈ V))
5 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
65sneqr 4798 . . . . . . . . 9 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
7 sneq 4596 . . . . . . . . 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 8932 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1→V)
11 f1f1orn 6795 . . . . . 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 8911 . . . . 5 ((𝑋 ∈ V ∧ (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
1412, 13mpdan 685 . . . 4 (𝑋 ∈ V → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
15 domen1 9063 . . . 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 22345 . . . . . . 7 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
18 simpr 485 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋) → 𝑥𝑋)
195snelpw 5402 . . . . . . . . . 10 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
2018, 19sylib 217 . . . . . . . . 9 ((𝑋 ∈ V ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
2120fmpttd 7063 . . . . . . . 8 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
2221frnd 6676 . . . . . . 7 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
23 elpwi 4567 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
2423ad2antrl 726 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑦𝑋)
25 simprr 771 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑦)
2624, 25sseldd 3945 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑋)
27 eqidd 2737 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} = {𝑧})
28 sneq 4596 . . . . . . . . . . . 12 (𝑥 = 𝑧 → {𝑥} = {𝑧})
2928rspceeqv 3595 . . . . . . . . . . 11 ((𝑧𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥𝑋 {𝑧} = {𝑥})
3026, 27, 29syl2anc 584 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑥𝑋 {𝑧} = {𝑥})
31 vsnex 5386 . . . . . . . . . . 11 {𝑧} ∈ V
32 eqid 2736 . . . . . . . . . . . 12 (𝑥𝑋 ↦ {𝑥}) = (𝑥𝑋 ↦ {𝑥})
3332elrnmpt 5911 . . . . . . . . . . 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 4623 . . . . . . . . . 10 𝑧 ∈ {𝑧}
3736a1i 11 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧 ∈ {𝑧})
3825snssd 4769 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ⊆ 𝑦)
39 eleq2 2826 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑧𝑤𝑧 ∈ {𝑧}))
40 sseq1 3969 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑤𝑦 ↔ {𝑧} ⊆ 𝑦))
4139, 40anbi12d 631 . . . . . . . . . 10 (𝑤 = {𝑧} → ((𝑧𝑤𝑤𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
4241rspcev 3581 . . . . . . . . 9 (({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4335, 37, 38, 42syl12anc 835 . . . . . . . 8 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4443ralrimivva 3197 . . . . . . 7 (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
45 basgen2 22339 . . . . . . 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 1371 . . . . . 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 2838 . . . . . . 7 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
49 tgclb 22320 . . . . . . 7 (ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
5048, 49sylibr 233 . . . . . 6 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases)
51 2ndci 22799 . . . . . 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 2839 . . . 4 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2ndω)
54 is2ndc 22797 . . . . . 6 (𝒫 𝑋 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
55 vex 3449 . . . . . . . . 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 776 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
5957, 58eleqtrrd 2841 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ (topGen‘𝑏))
60 vsnid 4623 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
61 tg2 22315 . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . 15 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑥𝑦)
6463snssd 4769 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
65 simprrr 780 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
6664, 65eqssd 3961 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
67 simprl 769 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦𝑏)
6866, 67eqeltrd 2838 . . . . . . . . . . . 12 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
6962, 68rexlimddv 3158 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝑏)
7069fmpttd 7063 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥𝑋 ↦ {𝑥}):𝑋𝑏)
7170frnd 6676 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏)
72 ssdomg 8940 . . . . . . . . 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 769 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
75 domtr 8947 . . . . . . . 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 3154 . . . . . 6 (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
7854, 77biimtrid 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 799 . . 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 379 1 (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  ran crn 5634  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  ωcom 7802  cen 8880  cdom 8881  topGenctg 17319  Topctop 22242  TopBasesctb 22295  2ndωc2ndc 22789
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8648  df-en 8884  df-dom 8885  df-topgen 17325  df-top 22243  df-bases 22296  df-2ndc 22791
This theorem is referenced by: (None)
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