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Theorem dis2ndc 21633
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 reldom 8227 . . 3 Rel ≼
21brrelex1i 5392 . 2 (𝑋 ≼ ω → 𝑋 ∈ V)
3 pwexr 7233 . 2 (𝒫 𝑋 ∈ 2nd𝜔 → 𝑋 ∈ V)
4 elex 3428 . . . . 5 (𝑋 ∈ V → 𝑋 ∈ V)
5 snex 5128 . . . . . . . 8 {𝑥} ∈ V
652a1i 12 . . . . . . 7 (𝑋 ∈ V → (𝑥𝑋 → {𝑥} ∈ V))
7 vex 3416 . . . . . . . . . 10 𝑥 ∈ V
87sneqr 4586 . . . . . . . . 9 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
9 sneq 4406 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
108, 9impbii 201 . . . . . . . 8 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
11102a1i 12 . . . . . . 7 (𝑋 ∈ V → ((𝑥𝑋𝑦𝑋) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)))
126, 11dom2lem 8261 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1→V)
13 f1f1orn 6388 . . . . . 6 ((𝑥𝑋 ↦ {𝑥}):𝑋1-1→V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥}))
1412, 13syl 17 . . . . 5 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥}))
15 f1oeng 8240 . . . . 5 ((𝑋 ∈ V ∧ (𝑥𝑋 ↦ {𝑥}):𝑋1-1-onto→ran (𝑥𝑋 ↦ {𝑥})) → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
164, 14, 15syl2anc 581 . . . 4 (𝑋 ∈ V → 𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}))
17 domen1 8370 . . . 4 (𝑋 ≈ ran (𝑥𝑋 ↦ {𝑥}) → (𝑋 ≼ ω ↔ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
1816, 17syl 17 . . 3 (𝑋 ∈ V → (𝑋 ≼ ω ↔ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
19 distop 21169 . . . . . . 7 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
20 simpr 479 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋) → 𝑥𝑋)
217snelpw 5133 . . . . . . . . . 10 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
2220, 21sylib 210 . . . . . . . . 9 ((𝑋 ∈ V ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
2322fmpttd 6633 . . . . . . . 8 (𝑋 ∈ V → (𝑥𝑋 ↦ {𝑥}):𝑋⟶𝒫 𝑋)
2423frnd 6284 . . . . . . 7 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋)
25 elpwi 4387 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
2625ad2antrl 721 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑦𝑋)
27 simprr 791 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑦)
2826, 27sseldd 3827 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧𝑋)
29 eqidd 2825 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} = {𝑧})
30 sneq 4406 . . . . . . . . . . . 12 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3130rspceeqv 3543 . . . . . . . . . . 11 ((𝑧𝑋 ∧ {𝑧} = {𝑧}) → ∃𝑥𝑋 {𝑧} = {𝑥})
3228, 29, 31syl2anc 581 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑥𝑋 {𝑧} = {𝑥})
33 snex 5128 . . . . . . . . . . 11 {𝑧} ∈ V
34 eqid 2824 . . . . . . . . . . . 12 (𝑥𝑋 ↦ {𝑥}) = (𝑥𝑋 ↦ {𝑥})
3534elrnmpt 5604 . . . . . . . . . . 11 ({𝑧} ∈ V → ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥}))
3633, 35ax-mp 5 . . . . . . . . . 10 ({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ↔ ∃𝑥𝑋 {𝑧} = {𝑥})
3732, 36sylibr 226 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}))
38 vsnid 4429 . . . . . . . . . 10 𝑧 ∈ {𝑧}
3938a1i 11 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → 𝑧 ∈ {𝑧})
4027snssd 4557 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → {𝑧} ⊆ 𝑦)
41 eleq2 2894 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑧𝑤𝑧 ∈ {𝑧}))
42 sseq1 3850 . . . . . . . . . . 11 (𝑤 = {𝑧} → (𝑤𝑦 ↔ {𝑧} ⊆ 𝑦))
4341, 42anbi12d 626 . . . . . . . . . 10 (𝑤 = {𝑧} → ((𝑧𝑤𝑤𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)))
4443rspcev 3525 . . . . . . . . 9 (({𝑧} ∈ ran (𝑥𝑋 ↦ {𝑥}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} ⊆ 𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4537, 39, 40, 44syl12anc 872 . . . . . . . 8 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋𝑧𝑦)) → ∃𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
4645ralrimivva 3179 . . . . . . 7 (𝑋 ∈ V → ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦))
47 basgen2 21163 . . . . . . 7 ((𝒫 𝑋 ∈ Top ∧ ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋𝑧𝑦𝑤 ∈ ran (𝑥𝑋 ↦ {𝑥})(𝑧𝑤𝑤𝑦)) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
4819, 24, 46, 47syl3anc 1496 . . . . . 6 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
4948adantr 474 . . . . 5 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) = 𝒫 𝑋)
5048, 19eqeltrd 2905 . . . . . . 7 (𝑋 ∈ V → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
51 tgclb 21144 . . . . . . 7 (ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ↔ (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ Top)
5250, 51sylibr 226 . . . . . 6 (𝑋 ∈ V → ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases)
53 2ndci 21621 . . . . . 6 ((ran (𝑥𝑋 ↦ {𝑥}) ∈ TopBases ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ 2nd𝜔)
5452, 53sylan 577 . . . . 5 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → (topGen‘ran (𝑥𝑋 ↦ {𝑥})) ∈ 2nd𝜔)
5549, 54eqeltrrd 2906 . . . 4 ((𝑋 ∈ V ∧ ran (𝑥𝑋 ↦ {𝑥}) ≼ ω) → 𝒫 𝑋 ∈ 2nd𝜔)
56 is2ndc 21619 . . . . . 6 (𝒫 𝑋 ∈ 2nd𝜔 ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋))
57 vex 3416 . . . . . . . . 9 𝑏 ∈ V
58 simpr 479 . . . . . . . . . . . . . . 15 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → 𝑥𝑋)
5958, 21sylib 210 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
60 simplrr 798 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → (topGen‘𝑏) = 𝒫 𝑋)
6159, 60eleqtrrd 2908 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ (topGen‘𝑏))
62 vsnid 4429 . . . . . . . . . . . . 13 𝑥 ∈ {𝑥}
63 tg2 21139 . . . . . . . . . . . . 13 (({𝑥} ∈ (topGen‘𝑏) ∧ 𝑥 ∈ {𝑥}) → ∃𝑦𝑏 (𝑥𝑦𝑦 ⊆ {𝑥}))
6461, 62, 63sylancl 582 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → ∃𝑦𝑏 (𝑥𝑦𝑦 ⊆ {𝑥}))
65 simprrl 801 . . . . . . . . . . . . . . 15 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑥𝑦)
6665snssd 4557 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ⊆ 𝑦)
67 simprrr 802 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦 ⊆ {𝑥})
6866, 67eqssd 3843 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} = 𝑦)
69 simprl 789 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → 𝑦𝑏)
7068, 69eqeltrd 2905 . . . . . . . . . . . 12 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) ∧ (𝑦𝑏 ∧ (𝑥𝑦𝑦 ⊆ {𝑥}))) → {𝑥} ∈ 𝑏)
7164, 70rexlimddv 3244 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) ∧ 𝑥𝑋) → {𝑥} ∈ 𝑏)
7271fmpttd 6633 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → (𝑥𝑋 ↦ {𝑥}):𝑋𝑏)
7372frnd 6284 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏)
74 ssdomg 8267 . . . . . . . . 9 (𝑏 ∈ V → (ran (𝑥𝑋 ↦ {𝑥}) ⊆ 𝑏 → ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏))
7557, 73, 74mpsyl 68 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏)
76 simprl 789 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → 𝑏 ≼ ω)
77 domtr 8274 . . . . . . . 8 ((ran (𝑥𝑋 ↦ {𝑥}) ≼ 𝑏𝑏 ≼ ω) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
7875, 76, 77syl2anc 581 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋)) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
7978rexlimdva2 3242 . . . . . 6 (𝑋 ∈ V → (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝒫 𝑋) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
8056, 79syl5bi 234 . . . . 5 (𝑋 ∈ V → (𝒫 𝑋 ∈ 2nd𝜔 → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω))
8180imp 397 . . . 4 ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2nd𝜔) → ran (𝑥𝑋 ↦ {𝑥}) ≼ ω)
8255, 81impbida 837 . . 3 (𝑋 ∈ V → (ran (𝑥𝑋 ↦ {𝑥}) ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔))
8318, 82bitrd 271 . 2 (𝑋 ∈ V → (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔))
842, 3, 83pm5.21nii 370 1 (𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3116  wrex 3117  Vcvv 3413  wss 3797  𝒫 cpw 4377  {csn 4396   class class class wbr 4872  cmpt 4951  ran crn 5342  1-1wf1 6119  1-1-ontowf1o 6121  cfv 6122  ωcom 7325  cen 8218  cdom 8219  topGenctg 16450  Topctop 21067  TopBasesctb 21119  2nd𝜔c2ndc 21611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-er 8008  df-en 8222  df-dom 8223  df-topgen 16456  df-top 21068  df-bases 21120  df-2ndc 21613
This theorem is referenced by: (None)
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