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Theorem tx2ndc 21668
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 21463 . 2 (𝑅 ∈ 2nd𝜔 ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2 is2ndc 21463 . 2 (𝑆 ∈ 2nd𝜔 ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3 reeanv 3295 . . 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 638 . . . . 5 (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5 txbasval 21623 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6 eqid 2806 . . . . . . . . . . 11 ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
76txval 21581 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
85, 7eqtrd 2840 . . . . . . . . 9 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
98adantr 468 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
106txbas 21584 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
1110adantr 468 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
12 omelon 8790 . . . . . . . . . . . 12 ω ∈ On
13 vex 3394 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
1413xpdom1 8298 . . . . . . . . . . . . . . 15 (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15 omex 8787 . . . . . . . . . . . . . . . 16 ω ∈ V
1615xpdom2 8294 . . . . . . . . . . . . . . 15 (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17 domtr 8245 . . . . . . . . . . . . . . 15 (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
1814, 16, 17syl2an 585 . . . . . . . . . . . . . 14 ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) → (𝑟 × 𝑠) ≼ (ω × ω))
1918adantl 469 . . . . . . . . . . . . 13 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
20 xpomen 9121 . . . . . . . . . . . . 13 (ω × ω) ≈ ω
21 domentr 8251 . . . . . . . . . . . . 13 (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
2219, 20, 21sylancl 576 . . . . . . . . . . . 12 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23 ondomen 9143 . . . . . . . . . . . 12 ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
2412, 22, 23sylancr 577 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25 eqid 2806 . . . . . . . . . . . . . 14 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
26 vex 3394 . . . . . . . . . . . . . . 15 𝑥 ∈ V
27 vex 3394 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2826, 27xpex 7192 . . . . . . . . . . . . . 14 (𝑥 × 𝑦) ∈ V
2925, 28fnmpt2i 7472 . . . . . . . . . . . . 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 6337 . . . . . . . . . . . 12 ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
3230, 31sylib 209 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
33 fodomnum 9163 . . . . . . . . . . 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 8245 . . . . . . . . . 10 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
3634, 22, 35syl2anc 575 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37 2ndci 21465 . . . . . . . . 9 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
3811, 36, 37syl2anc 575 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
399, 38eqeltrd 2885 . . . . . . 7 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔)
40 oveq12 6883 . . . . . . . 8 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
4140eleq1d 2870 . . . . . . 7 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔 ↔ (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4239, 41syl5ibcom 236 . . . . . 6 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4342expimpd 443 . . . . 5 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
444, 43syl5bi 233 . . . 4 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
4544rexlimivv 3224 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
463, 45sylbir 226 . 2 ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
471, 2, 46syl2anb 587 1 ((𝑅 ∈ 2nd𝜔 ∧ 𝑆 ∈ 2nd𝜔) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wrex 3097   class class class wbr 4844   × cxp 5309  dom cdm 5311  ran crn 5312  Oncon0 5936   Fn wfn 6096  ontowfo 6099  cfv 6101  (class class class)co 6874  cmpt2 6876  ωcom 7295  cen 8189  cdom 8190  cardccrd 9044  topGenctg 16303  TopBasesctb 20963  2nd𝜔c2ndc 21455   ×t ctx 21577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-oi 8654  df-card 9048  df-acn 9051  df-topgen 16309  df-bases 20964  df-2ndc 21457  df-tx 21579
This theorem is referenced by: (None)
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