Theorem xkopjcn 22258

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 22286.) (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 2821	. . . . . 6 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
2	1	xkotopon 22202	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
3	2	3adant3 1128	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4		xkopjcn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
5	4	topopn 21508	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
6	5	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ 𝑅)
7		fconst6g 6562	. . . . . . . 8 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
8	7	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9		pttop 22184	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
10	6, 8, 9	syl2anc 586	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11		eqid 2821	. . . . . . . . . 10 ⊢ ∪ 𝑆 = ∪ 𝑆
12	4, 11	cnf 21848	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋⟶∪ 𝑆)
13		uniexg 7460	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ V)
14	13	3ad2ant2 1130	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ∪ 𝑆 ∈ V)
15	14, 6	elmapd 8414	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (∪ 𝑆 ↑m 𝑋) ↔ 𝑓:𝑋⟶∪ 𝑆))
16	12, 15	syl5ibr 248	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ (∪ 𝑆 ↑m 𝑋)))
17	16	ssrdv 3972	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
18		simp2 1133	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ Top)
19		eqid 2821	. . . . . . . . 9 ⊢ (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
20	19, 11	ptuniconst 22200	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
21	6, 18, 20	syl2anc 586	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
22	17, 21	sseqtrd 4006	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆})))
23		eqid 2821	. . . . . . 7 ⊢ ∪ (∏t‘(𝑋 × {𝑆})) = ∪ (∏t‘(𝑋 × {𝑆}))
24	23	restuni 21764	. . . . . 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 586	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
26	25	fveq2d 6668	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
27	3, 26	eleqtrd 2915	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
28	4, 19	xkoptsub 22256	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
29	28	3adant3 1128	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
30		eqid 2821	. . . 4 ⊢ ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
31	30	cnss1 21878	. . 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 586	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆 ↑ko 𝑅) Cn 𝑆))
33	22	resmptd 5902	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)))
34		simp3 1134	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
35	23, 19	ptpjcn 22213	. . . . . 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 1367	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37		fvconst2g 6958	. . . . . . 7 ⊢ ((𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38	37	3adant1 1126	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
39	38	oveq2d 7166	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
40	36, 39	eleqtrd 2915	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
41	23	cnrest 21887	. . . 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 586	. . 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 2914	. 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 3967	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  {csn 4560  ∪ cuni 4831   ↦ cmpt 5138   × cxp 5547   ↾ cres 5551  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ↑_m cmap 8400   ↾_t crest 16688  ∏_tcpt 16706  Topctop 21495  TopOnctopon 21512   Cn ccn 21826   ↑_ko cxko 22163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-rest 16690  df-topgen 16711  df-pt 16712  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-cmp 21989  df-xko 22165

This theorem is referenced by:  cnmptkp  22282  xkofvcn  22286