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Theorem tx2ndc 23002
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 22797 . 2 (𝑅 ∈ 2ndω ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2 is2ndc 22797 . 2 (𝑆 ∈ 2ndω ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3 reeanv 3217 . . 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 654 . . . . 5 (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5 txbasval 22957 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6 eqid 2736 . . . . . . . . . . 11 ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
76txval 22915 . . . . . . . . . 10 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
85, 7eqtrd 2776 . . . . . . . . 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 22918 . . . . . . . . . 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 9582 . . . . . . . . . . . 12 ω ∈ On
13 vex 3449 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
1413xpdom1 9015 . . . . . . . . . . . . . . 15 (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15 omex 9579 . . . . . . . . . . . . . . . 16 ω ∈ V
1615xpdom2 9011 . . . . . . . . . . . . . . 15 (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17 domtr 8947 . . . . . . . . . . . . . . 15 (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
1814, 16, 17syl2an 596 . . . . . . . . . . . . . 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 9951 . . . . . . . . . . . . 13 (ω × ω) ≈ ω
21 domentr 8953 . . . . . . . . . . . . 13 (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
2219, 20, 21sylancl 586 . . . . . . . . . . . 12 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23 ondomen 9973 . . . . . . . . . . . 12 ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
2412, 22, 23sylancr 587 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25 eqid 2736 . . . . . . . . . . . . . 14 (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))
26 vex 3449 . . . . . . . . . . . . . . 15 𝑥 ∈ V
27 vex 3449 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2826, 27xpex 7687 . . . . . . . . . . . . . 14 (𝑥 × 𝑦) ∈ V
2925, 28fnmpoi 8002 . . . . . . . . . . . . 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 6762 . . . . . . . . . . . 12 ((𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
3230, 31sylib 217 . . . . . . . . . . 11 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)))
33 fodomnum 9993 . . . . . . . . . . 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 8947 . . . . . . . . . 10 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
3634, 22, 35syl2anc 584 . . . . . . . . 9 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37 2ndci 22799 . . . . . . . . 9 ((ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
3811, 36, 37syl2anc 584 . . . . . . . 8 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
399, 38eqeltrd 2838 . . . . . . 7 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω)
40 oveq12 7366 . . . . . . . 8 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
4140eleq1d 2822 . . . . . . 7 (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω ↔ (𝑅 ×t 𝑆) ∈ 2ndω))
4239, 41syl5ibcom 244 . . . . . 6 (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2ndω))
4342expimpd 454 . . . . 5 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
444, 43biimtrid 241 . . . 4 ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
4544rexlimivv 3196 . . 3 (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
463, 45sylbir 234 . 2 ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
471, 2, 46syl2anb 598 1 ((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3073   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  Oncon0 6317   Fn wfn 6491  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  ωcom 7802  cen 8880  cdom 8881  cardccrd 9871  topGenctg 17319  TopBasesctb 22295  2ndωc2ndc 22789   ×t ctx 22911
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-card 9875  df-acn 9878  df-topgen 17325  df-bases 22296  df-2ndc 22791  df-tx 22913
This theorem is referenced by: (None)
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