Theorem xkopjcn 23782

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 23810.) (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 2769	. . . . . 6 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
2	1	xkotopon 23726	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
3	2	3adant3 1148	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4		xkopjcn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
5	4	topopn 23032	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
6	5	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ 𝑅)
7		fconst6g 6768	. . . . . . . 8 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
8	7	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9		pttop 23708	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
10	6, 8, 9	syl2anc 595	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11		eqid 2769	. . . . . . . . . 10 ⊢ ∪ 𝑆 = ∪ 𝑆
12	4, 11	cnf 23372	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋⟶∪ 𝑆)
13		uniexg 7739	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ V)
14	13	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ∪ 𝑆 ∈ V)
15	14, 6	elmapd 8837	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑓:𝑋⟶∪ 𝑆))
16	12, 15	imbitrrid 249	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋)))
17	16	ssrdv 3951	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
18		simp2 1153	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ Top)
19		eqid 2769	. . . . . . . . 9 ⊢ (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
20	19, 11	ptuniconst 23724	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
21	6, 18, 20	syl2anc 595	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
22	17, 21	sseqtrd 3981	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆})))
23		eqid 2769	. . . . . . 7 ⊢ ∪ (∏t‘(𝑋 × {𝑆})) = ∪ (∏t‘(𝑋 × {𝑆}))
24	23	restuni 23288	. . . . . 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 595	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
26	25	fveq2d 6886	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
27	3, 26	eleqtrd 2871	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
28	4, 19	xkoptsub 23780	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
29	28	3adant3 1148	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
30		eqid 2769	. . . 4 ⊢ ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
31	30	cnss1 23402	. . 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 595	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆))
33	22	resmptd 6043	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)))
34		simp3 1154	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
35	23, 19	ptpjcn 23737	. . . . . 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 1396	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37		fvconst2g 7201	. . . . . . 7 ⊢ ((𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38	37	3adant1 1146	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
39	38	oveq2d 7427	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
40	36, 39	eleqtrd 2871	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
41	23	cnrest 23411	. . . 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 595	. . 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 2870	. 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 3946	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  {csn 4594  ∪ cuni 4876   ↦ cmpt 5196   × cxp 5660   ↾ cres 5664  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8824   ↾_t crest 17473  ∏_tcpt 17491  Topctop 23019  TopOnctopon 23036   Cn ccn 23350   ↑_ko cxko 23687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-1o 8453  df-2o 8454  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-pt 17497  df-top 23020  df-topon 23037  df-bases 23072  df-cn 23353  df-cmp 23513  df-xko 23689

This theorem is referenced by:  cnmptkp  23806  xkofvcn  23810