Theorem tx2ndc 23566

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 23361	. 2 ⊢ (𝑅 ∈ 2ndω ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2		is2ndc 23361	. 2 ⊢ (𝑆 ∈ 2ndω ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3		reeanv 3204	. . 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 656	. . . . 5 ⊢ (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5		txbasval 23521	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6		eqid 2731	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
7	6	txval 23479	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
8	5, 7	eqtrd 2766	. . . . . . . . 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 23482	. . . . . . . . . 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 9536	. . . . . . . . . . . 12 ⊢ ω ∈ On
13		vex 3440	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
14	13	xpdom1 8989	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15		omex 9533	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ V
16	15	xpdom2 8985	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17		domtr 8929	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
18	14, 16, 17	syl2an 596	. . . . . . . . . . . . . 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 9906	. . . . . . . . . . . . 13 ⊢ (ω × ω) ≈ ω
21		domentr 8935	. . . . . . . . . . . . 13 ⊢ (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
22	19, 20, 21	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23		ondomen 9928	. . . . . . . . . . . 12 ⊢ ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
24	12, 22, 23	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
26		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
28	26, 27	xpex 7686	. . . . . . . . . . . . . 14 ⊢ (𝑥 × 𝑦) ∈ V
29	25, 28	fnmpoi 8002	. . . . . . . . . . . . 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 6741	. . . . . . . . . . . 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 218	. . . . . . . . . . 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 9948	. . . . . . . . . . 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 65	. . . . . . . . . 10 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ (𝑟 × 𝑠))
35		domtr 8929	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37		2ndci 23363	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
39	9, 38	eqeltrd 2831	. . . . . . 7 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω)
40		oveq12 7355	. . . . . . . 8 ⊢ (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
41	40	eleq1d 2816	. . . . . . 7 ⊢ (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω ↔ (𝑅 ×t 𝑆) ∈ 2ndω))
42	39, 41	syl5ibcom 245	. . . . . 6 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → (𝑅 ×t 𝑆) ∈ 2ndω))
43	42	expimpd 453	. . . . 5 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
44	4, 43	biimtrid 242	. . . 4 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω))
45	44	rexlimivv 3174	. . 3 ⊢ (∃𝑟 ∈ TopBases ∃𝑠 ∈ TopBases ((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
46	3, 45	sylbir 235	. 2 ⊢ ((∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) → (𝑅 ×t 𝑆) ∈ 2ndω)
47	1, 2, 46	syl2anb 598	1 ⊢ ((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   class class class wbr 5089   × cxp 5612  dom cdm 5614  ran crn 5615  Oncon0 6306   Fn wfn 6476  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ωcom 7796   ≈ cen 8866   ≼ cdom 8867  cardccrd 9828  topGenctg 17341  TopBasesctb 22860  2^ndωc2ndc 23353   ×_t ctx 23475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-card 9832  df-acn 9835  df-topgen 17347  df-bases 22861  df-2ndc 23355  df-tx 23477

This theorem is referenced by: (None)