Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tx2ndc Structured version   Visualization version   GIF version

Theorem tx2ndc 22254
 Description: The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
tx2ndc ((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)

Proof of Theorem tx2ndc
Dummy variables 𝑠 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 22049 . 2 (𝑅 ∈ 2ndω ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2 is2ndc 22049 . 2 (𝑆 ∈ 2ndω ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3 reeanv 3348 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ (∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)))
4 an4 655 . . . . 5 (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5 txbasval 22209 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6 eqid 2822 . . . . . . . . . . 11 ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
76txval 22167 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
85, 7eqtrd 2857 . . . . . . . . 9 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
98adantr 484 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
106txbas 22170 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
1110adantr 484 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
12 omelon 9097 . . . . . . . . . . . 12 ω ∈ On
13 vex 3472 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
1413xpdom1 8603 . . . . . . . . . . . . . . 15 (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15 omex 9094 . . . . . . . . . . . . . . . 16 ω ∈ V
1615xpdom2 8599 . . . . . . . . . . . . . . 15 (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17 domtr 8549 . . . . . . . . . . . . . . 15 (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
1814, 16, 17syl2an 598 . . . . . . . . . . . . . 14 ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) → (𝑟 × 𝑠) ≼ (ω × ω))
1918adantl 485 . . . . . . . . . . . . 13 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
20 xpomen 9430 . . . . . . . . . . . . 13 (ω × ω) ≈ ω
21 domentr 8555 . . . . . . . . . . . . 13 (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
2219, 20, 21sylancl 589 . . . . . . . . . . . 12 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23 ondomen 9452 . . . . . . . . . . . 12 ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
2412, 22, 23sylancr 590 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25 eqid 2822 . . . . . . . . . . . . . 14 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
26 vex 3472 . . . . . . . . . . . . . . 15 𝑥 ∈ V
27 vex 3472 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2826, 27xpex 7461 . . . . . . . . . . . . . 14 (𝑥 × 𝑦) ∈ V
2925, 28fnmpoi 7754 . . . . . . . . . . . . 13 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠)
3029a1i 11 . . . . . . . . . . . 12 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠))
31 dffn4 6578 . . . . . . . . . . . 12 ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
3230, 31sylib 221 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
33 fodomnum 9472 . . . . . . . . . . 11 ((𝑟 × 𝑠) ∈ dom card → ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠)))
3424, 32, 33sylc 65 . . . . . . . . . 10 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠))
35 domtr 8549 . . . . . . . . . 10 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
3634, 22, 35syl2anc 587 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37 2ndci 22051 . . . . . . . . 9 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
3811, 36, 37syl2anc 587 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
399, 38eqeltrd 2914 . . . . . . 7 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω)
40 oveq12 7149 . . . . . . . 8 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
4140eleq1d 2898 . . . . . . 7 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω ↔ (𝑅 ×t 𝑆) ∈ 2ndω))
4239, 41syl5ibcom 248 . . . . . 6 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2ndω))
4342expimpd 457 . . . . 5 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
444, 43syl5bi 245 . . . 4 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
4544rexlimivv 3278 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
463, 45sylbir 238 . 2 ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
471, 2, 46syl2anb 600 1 ((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∃wrex 3131   class class class wbr 5042   × cxp 5530  dom cdm 5532  ran crn 5533  Oncon0 6169   Fn wfn 6329  –onto→wfo 6332  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142  ωcom 7565   ≈ cen 8493   ≼ cdom 8494  cardccrd 9352  topGenctg 16702  TopBasesctb 21548  2ndωc2ndc 22041   ×t ctx 22163 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-card 9356  df-acn 9359  df-topgen 16708  df-bases 21549  df-2ndc 22043  df-tx 22165 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator