Theorem tx2ndc 23604

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 23399	. 2 ⊢ (𝑅 ∈ 2ndω ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2		is2ndc 23399	. 2 ⊢ (𝑆 ∈ 2ndω ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3		reeanv 3207	. . 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 657	. . . . 5 ⊢ (((𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅) ∧ (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆)) ↔ ((𝑟 ≼ ω ∧ 𝑠 ≼ ω) ∧ ((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆)))
5		txbasval 23559	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑟 ×t 𝑠))
6		eqid 2735	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
7	6	txval 23517	. . . . . . . . . 10 ⊢ ((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) → (𝑟 ×t 𝑠) = (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
8	5, 7	eqtrd 2770	. . . . . . . . 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 23520	. . . . . . . . . 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 9556	. . . . . . . . . . . 12 ⊢ ω ∈ On
13		vex 3431	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
14	13	xpdom1 9003	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15		omex 9553	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ V
16	15	xpdom2 8999	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17		domtr 8943	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 × 𝑠) ≼ (ω × 𝑠) ∧ (ω × 𝑠) ≼ (ω × ω)) → (𝑟 × 𝑠) ≼ (ω × ω))
18	14, 16, 17	syl2an 597	. . . . . . . . . . . . . 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 9926	. . . . . . . . . . . . 13 ⊢ (ω × ω) ≈ ω
21		domentr 8949	. . . . . . . . . . . . 13 ⊢ (((𝑟 × 𝑠) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝑟 × 𝑠) ≼ ω)
22	19, 20, 21	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ≼ ω)
23		ondomen 9948	. . . . . . . . . . . 12 ⊢ ((ω ∈ On ∧ (𝑟 × 𝑠) ≼ ω) → (𝑟 × 𝑠) ∈ dom card)
24	12, 22, 23	sylancr 588	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (𝑟 × 𝑠) ∈ dom card)
25		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
26		vex 3431	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27		vex 3431	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
28	26, 27	xpex 7696	. . . . . . . . . . . . . 14 ⊢ (𝑥 × 𝑦) ∈ V
29	25, 28	fnmpoi 8012	. . . . . . . . . . . . 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 6747	. . . . . . . . . . . 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 9968	. . . . . . . . . . 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 8943	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) ≼ ω)
37		2ndci 23401	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))) ∈ 2ndω)
39	9, 38	eqeltrd 2835	. . . . . . 7 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω)
40		oveq12 7365	. . . . . . . 8 ⊢ (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
41	40	eleq1d 2820	. . . . . . 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 3177	. . 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 599	1 ⊢ ((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   class class class wbr 5074   × cxp 5618  dom cdm 5620  ran crn 5621  Oncon0 6312   Fn wfn 6482  –onto→wfo 6485  ‘cfv 6487  (class class class)co 7356   ∈ cmpo 7358  ωcom 7806   ≈ cen 8879   ≼ cdom 8880  cardccrd 9848  topGenctg 17389  TopBasesctb 22898  2^ndωc2ndc 23391   ×_t ctx 23513

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8632  df-map 8764  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-oi 9414  df-card 9852  df-acn 9855  df-topgen 17395  df-bases 22899  df-2ndc 23393  df-tx 23515

This theorem is referenced by: (None)