Theorem xkopjcn 23718

Description: Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both 𝑓 and 𝐴 as a function on (𝑆 ↑_ko 𝑅) ×_t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 23746.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypothesis

Ref	Expression
xkopjcn.1	⊢ 𝑋 = ∪ 𝑅

Assertion

Ref	Expression
xkopjcn	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   𝑓,𝑋

Proof of Theorem xkopjcn

Step	Hyp	Ref	Expression
1		eqid 2764	. . . . . 6 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
2	1	xkotopon 23662	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
3	2	3adant3 1146	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4		xkopjcn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
5	4	topopn 22968	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
6	5	3ad2ant1 1147	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ 𝑅)
7		fconst6g 6755	. . . . . . . 8 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
8	7	3ad2ant2 1148	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9		pttop 23644	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
10	6, 8, 9	syl2anc 593	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11		eqid 2764	. . . . . . . . . 10 ⊢ ∪ 𝑆 = ∪ 𝑆
12	4, 11	cnf 23308	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋⟶∪ 𝑆)
13		uniexg 7725	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ V)
14	13	3ad2ant2 1148	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ∪ 𝑆 ∈ V)
15	14, 6	elmapd 8823	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑓:𝑋⟶∪ 𝑆))
16	12, 15	imbitrrid 248	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋)))
17	16	ssrdv 3944	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
18		simp2 1151	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ Top)
19		eqid 2764	. . . . . . . . 9 ⊢ (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
20	19, 11	ptuniconst 23660	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
21	6, 18, 20	syl2anc 593	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
22	17, 21	sseqtrd 3974	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆})))
23		eqid 2764	. . . . . . 7 ⊢ ∪ (∏t‘(𝑋 × {𝑆})) = ∪ (∏t‘(𝑋 × {𝑆}))
24	23	restuni 23224	. . . . . 6 ⊢ (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
25	10, 22, 24	syl2anc 593	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
26	25	fveq2d 6873	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
27	3, 26	eleqtrd 2866	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
28	4, 19	xkoptsub 23716	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
29	28	3adant3 1146	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
30		eqid 2764	. . . 4 ⊢ ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
31	30	cnss1 23338	. . 3 ⊢ (((𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) ∧ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅)) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆))
32	27, 29, 31	syl2anc 593	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆))
33	22	resmptd 6031	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)))
34		simp3 1152	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
35	23, 19	ptpjcn 23673	. . . . . 6 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
36	6, 8, 34, 35	syl3anc 1392	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37		fvconst2g 7188	. . . . . . 7 ⊢ ((𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38	37	3adant1 1144	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
39	38	oveq2d 7414	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
40	36, 39	eleqtrd 2866	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
41	23	cnrest 23347	. . . 4 ⊢ (((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆) ∧ (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆}))) → ((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
42	40, 22, 41	syl2anc 593	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
43	33, 42	eqeltrrd 2865	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
44	32, 43	sseldd 3939	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ⊆ wss 3906  {csn 4584  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5647   ↾ cres 5651  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ↑_m cmap 8810   ↾_t crest 17451  ∏_tcpt 17469  Topctop 22955  TopOnctopon 22972   Cn ccn 23286   ↑_ko cxko 23623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-1o 8439  df-2o 8440  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-fin 8933  df-fi 9359  df-rest 17453  df-topgen 17474  df-pt 17475  df-top 22956  df-topon 22973  df-bases 23008  df-cn 23289  df-cmp 23449  df-xko 23625

This theorem is referenced by:  cnmptkp  23742  xkofvcn  23746