Theorem tx2ndc 23589

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 23384	. 2 ⊢ (𝑅 ∈ 2ndω ↔ ∃𝑟 ∈ TopBases (𝑟 ≼ ω ∧ (topGen‘𝑟) = 𝑅))
2		is2ndc 23384	. 2 ⊢ (𝑆 ∈ 2ndω ↔ ∃𝑠 ∈ TopBases (𝑠 ≼ ω ∧ (topGen‘𝑠) = 𝑆))
3		reeanv 3213	. . 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 23544	. . . . . . . . . 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 23502	. . . . . . . . . 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 23505	. . . . . . . . . 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 9660	. . . . . . . . . . . 12 ⊢ ω ∈ On
13		vex 3463	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
14	13	xpdom1 9085	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ≼ ω → (𝑟 × 𝑠) ≼ (ω × 𝑠))
15		omex 9657	. . . . . . . . . . . . . . . 16 ⊢ ω ∈ V
16	15	xpdom2 9081	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ≼ ω → (ω × 𝑠) ≼ (ω × ω))
17		domtr 9021	. . . . . . . . . . . . . . 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 10029	. . . . . . . . . . . . 13 ⊢ (ω × ω) ≈ ω
21		domentr 9027	. . . . . . . . . . . . 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 10051	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))
26		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
28	26, 27	xpex 7747	. . . . . . . . . . . . . 14 ⊢ (𝑥 × 𝑦) ∈ V
29	25, 28	fnmpoi 8069	. . . . . . . . . . . . 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 6796	. . . . . . . . . . . 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 10071	. . . . . . . . . . 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 9021	. . . . . . . . . 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 23386	. . . . . . . . 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 2834	. . . . . . 7 ⊢ (((𝑟 ∈ TopBases ∧ 𝑠 ∈ TopBases) ∧ (𝑟 ≼ ω ∧ 𝑠 ≼ ω)) → ((topGen‘𝑟) ×t (topGen‘𝑠)) ∈ 2ndω)
40		oveq12 7414	. . . . . . . 8 ⊢ (((topGen‘𝑟) = 𝑅 ∧ (topGen‘𝑠) = 𝑆) → ((topGen‘𝑟) ×t (topGen‘𝑠)) = (𝑅 ×t 𝑆))
41	40	eleq1d 2819	. . . . . . 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 3186	. . 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 1540   ∈ wcel 2108  ∃wrex 3060   class class class wbr 5119   × cxp 5652  dom cdm 5654  ran crn 5655  Oncon0 6352   Fn wfn 6526  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861   ≈ cen 8956   ≼ cdom 8957  cardccrd 9949  topGenctg 17451  TopBasesctb 22883  2^ndωc2ndc 23376   ×_t ctx 23498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-oi 9524  df-card 9953  df-acn 9956  df-topgen 17457  df-bases 22884  df-2ndc 23378  df-tx 23500

This theorem is referenced by: (None)