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Theorem tx2ndc 21364
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 21159 . 2 (𝑅 ∈ 2nd𝜔 ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2 is2ndc 21159 . 2 (𝑆 ∈ 2nd𝜔 ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3 reeanv 3097 . . 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 864 . . . . 5 (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5 txbasval 21319 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6 eqid 2621 . . . . . . . . . . 11 ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
76txval 21277 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
85, 7eqtrd 2655 . . . . . . . . 9 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
98adantr 481 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
106txbas 21280 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
1110adantr 481 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
12 omelon 8487 . . . . . . . . . . . 12 ω ∈ On
13 vex 3189 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
1413xpdom1 8003 . . . . . . . . . . . . . . 15 (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15 omex 8484 . . . . . . . . . . . . . . . 16 ω ∈ V
1615xpdom2 7999 . . . . . . . . . . . . . . 15 (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17 domtr 7953 . . . . . . . . . . . . . . 15 (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
1814, 16, 17syl2an 494 . . . . . . . . . . . . . 14 ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) → (𝑟 × 𝑠) ≼ (ω × ω))
1918adantl 482 . . . . . . . . . . . . 13 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
20 xpomen 8782 . . . . . . . . . . . . 13 (ω × ω) ≈ ω
21 domentr 7959 . . . . . . . . . . . . 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 8804 . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . 14 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
26 vex 3189 . . . . . . . . . . . . . . 15 𝑥 ∈ V
27 vex 3189 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2826, 27xpex 6915 . . . . . . . . . . . . . 14 (𝑥 × 𝑦) ∈ V
2925, 28fnmpt2i 7184 . . . . . . . . . . . . 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 6078 . . . . . . . . . . . 12 ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
3230, 31sylib 208 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
33 fodomnum 8824 . . . . . . . . . . 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 7953 . . . . . . . . . 10 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
3634, 22, 35syl2anc 692 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37 2ndci 21161 . . . . . . . . 9 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
3811, 36, 37syl2anc 692 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
399, 38eqeltrd 2698 . . . . . . 7 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔)
40 oveq12 6613 . . . . . . . 8 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
4140eleq1d 2683 . . . . . . 7 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔 ↔ (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4239, 41syl5ibcom 235 . . . . . 6 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4342expimpd 628 . . . . 5 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
444, 43syl5bi 232 . . . 4 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4544rexlimivv 3029 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
463, 45sylbir 225 . 2 ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
471, 2, 46syl2anb 496 1 ((𝑅 ∈ 2nd𝜔 ∧ 𝑆 ∈ 2nd𝜔) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2908   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075  Oncon0 5682   Fn wfn 5842  ontowfo 5845  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  cen 7896  cdom 7897  cardccrd 8705  topGenctg 16019  TopBasesctb 20620  2nd𝜔c2ndc 21151   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-card 8709  df-acn 8712  df-topgen 16025  df-bases 20622  df-2ndc 21153  df-tx 21275
This theorem is referenced by: (None)
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