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.

Step	Hyp	Ref	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 (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
7	6	txval 21177	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
8	5, 7	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
9	8	adantr 480	. . . . . . . 8 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
10	6	txbas 21180	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
11	10	adantr 480	. . . . . . . . 9 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases)
12		omelon 8426	. . . . . . . . . . . 12 ⊢ ω ∈ On
13		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
14	13	xpdom1 7944	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15		omex 8423	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ V
16	15	xpdom2 7940	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17		domtr 7895	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
18	14, 16, 17	syl2an 493	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) → (𝑟 × 𝑠) ≼ (ω × ω))
19	18	adantl 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)
22	19, 20, 21	sylancl 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)
24	12, 22, 23	sylancr 694	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
26		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
28	26, 27	xpex 6860	. . . . . . . . . . . . . 14 ⊢ (𝑥 × 𝑦) ∈ V
29	25, 28	fnmpt2i 7128	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠)
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠))
31		dffn4 6034	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) Fn (𝑟 × 𝑠) ↔ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))
32	30, 31	sylib 207	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))
33		fodomnum 8763	. . . . . . . . . . 11 ⊢ ((𝑟 × 𝑠) ∈ dom card → ((𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)):(𝑟 × 𝑠)–onto→ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠)))
34	24, 32, 33	sylc 63	. . . . . . . . . 10 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠))
35		domtr 7895	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ≼ ω) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
36	34, 22, 35	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37		2ndci 21061	. . . . . . . . 9 ⊢ ((ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ∈ TopBases ∧ ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ ω) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
38	11, 36, 37	syl2anc 691	. . . . . . . 8 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) ∈ 2nd𝜔)
39	9, 38	eqeltrd 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))
41	40	eleq1d 2672	. . . . . . 7 ⊢ (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2nd𝜔 ↔ (𝑅 ×t 𝑆) ∈ 2nd𝜔))
42	39, 41	syl5ibcom 234	. . . . . 6 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
43	42	expimpd 627	. . . . 5 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
44	4, 43	syl5bi 231	. . . 4 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔))
45	44	rexlimivv 3018	. . 3 ⊢ (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
46	3, 45	sylbir 224	. 2 ⊢ ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)
47	1, 2, 46	syl2anb 495	1 ⊢ ((𝑅 ∈ 2nd𝜔 ∧ 𝑆 ∈ 2nd𝜔) → (𝑅 ×t 𝑆) ∈ 2nd𝜔)

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  2^nd𝜔c2ndc 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)