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

Theorem tx2ndc 21264
 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 21059 . 2 (𝑅 ∈ 2nd𝜔 ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2 is2ndc 21059 . 2 (𝑆 ∈ 2nd𝜔 ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3 reeanv 3086 . . 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 861 . . . . 5 (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5 txbasval 21219 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6 eqid 2610 . . . . . . . . . . 11 ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
76txval 21177 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
85, 7eqtrd 2644 . . . . . . . . 9 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
98adantr 480 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
106txbas 21180 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
1110adantr 480 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
12 omelon 8426 . . . . . . . . . . . 12 ω ∈ On
13 vex 3176 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
1413xpdom1 7944 . . . . . . . . . . . . . . 15 (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15 omex 8423 . . . . . . . . . . . . . . . 16 ω ∈ V
1615xpdom2 7940 . . . . . . . . . . . . . . 15 (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17 domtr 7895 . . . . . . . . . . . . . . 15 (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
1814, 16, 17syl2an 493 . . . . . . . . . . . . . 14 ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) → (𝑟 × 𝑠) ≼ (ω × ω))
1918adantl 481 . . . . . . . . . . . . 13 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
20 xpomen 8721 . . . . . . . . . . . . 13 (ω × ω) ≈ ω
21 domentr 7901 . . . . . . . . . . . . 13 (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
2219, 20, 21sylancl 693 . . . . . . . . . . . 12 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23 ondomen 8743 . . . . . . . . . . . 12 ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
2412, 22, 23sylancr 694 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25 eqid 2610 . . . . . . . . . . . . . 14 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
26 vex 3176 . . . . . . . . . . . . . . 15 𝑥 ∈ V
27 vex 3176 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2826, 27xpex 6860 . . . . . . . . . . . . . 14 (𝑥 × 𝑦) ∈ V
2925, 28fnmpt2i 7128 . . . . . . . . . . . . 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 6034 . . . . . . . . . . . 12 ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
3230, 31sylib 207 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
33 fodomnum 8763 . . . . . . . . . . 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 63 . . . . . . . . . 10 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠))
35 domtr 7895 . . . . . . . . . 10 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
3634, 22, 35syl2anc 691 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37 2ndci 21061 . . . . . . . . 9 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
3811, 36, 37syl2anc 691 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
399, 38eqeltrd 2688 . . . . . . 7 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔)
40 oveq12 6558 . . . . . . . 8 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
4140eleq1d 2672 . . . . . . 7 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔 ↔ (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4239, 41syl5ibcom 234 . . . . . 6 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4342expimpd 627 . . . . 5 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
444, 43syl5bi 231 . . . 4 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4544rexlimivv 3018 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
463, 45sylbir 224 . 2 ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
471, 2, 46syl2anb 495 1 ((𝑅 ∈ 2nd𝜔 ∧ 𝑆 ∈ 2nd𝜔) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039  Oncon0 5640   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ≈ cen 7838   ≼ cdom 7839  cardccrd 8644  topGenctg 15921  TopBasesctb 20520  2nd𝜔c2ndc 21051   ×t ctx 21173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-acn 8651  df-topgen 15927  df-bases 20522  df-2ndc 21053  df-tx 21175 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator