Theorem xkopjcn 24002

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 24030.) (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 2763	. . . . . 6 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
2	1	xkotopon 23946	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
3	2	3adant3 1150	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4		xkopjcn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
5	4	topopn 23251	. . . . . . . 8 ⊢ (𝑅 ∈ Top → 𝑋 ∈ 𝑅)
6	5	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑋 ∈ 𝑅)
7		fconst6g 6790	. . . . . . . 8 ⊢ (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
8	7	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9		pttop 23928	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
10	6, 8, 9	syl2anc 596	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝑆 = ∪ 𝑆
12	4, 11	cnf 23591	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋⟶∪ 𝑆)
13		uniexg 7778	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ V)
14	13	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ∪ 𝑆 ∈ V)
15	14, 6	elmapd 8885	. . . . . . . . 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 3919	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ (∪ 𝑆 ↑m 𝑋))
18		simp2 1155	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ Top)
19		eqid 2763	. . . . . . . . 9 ⊢ (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
20	19, 11	ptuniconst 23944	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑆 ∈ Top) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
21	6, 18, 20	syl2anc 596	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∪ 𝑆 ↑m 𝑋) = ∪ (∏t‘(𝑋 × {𝑆})))
22	17, 21	sseqtrd 3949	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) ⊆ ∪ (∏t‘(𝑋 × {𝑆})))
23		eqid 2763	. . . . . . 7 ⊢ ∪ (∏t‘(𝑋 × {𝑆})) = ∪ (∏t‘(𝑋 × {𝑆}))
24	23	restuni 23507	. . . . . 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 596	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑅 Cn 𝑆) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
26	25	fveq2d 6909	. . . 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 24000	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
29	28	3adant3 1150	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
30		eqid 2763	. . . 4 ⊢ ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ∪ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
31	30	cnss1 23621	. . 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 596	. 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 1156	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
35	23, 19	ptpjcn 23957	. . . . . 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 1399	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ∪ (∏t‘(𝑋 × {𝑆})) ↦ (𝑓‘𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37		fvconst2g 7228	. . . . . . 7 ⊢ ((𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38	37	3adant1 1148	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
39	38	oveq2d 7456	. . . . 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 23630	. . . 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 596	. . 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 3914	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189   × cxp 5653   ↾ cres 5657  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872   ↾_t crest 17617  ∏_tcpt 17635  Topctop 23238  TopOnctopon 23255   Cn ccn 23569   ↑_ko cxko 23907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-1o 8501  df-2o 8502  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-fin 9002  df-fi 9427  df-rest 17619  df-topgen 17640  df-pt 17641  df-top 23239  df-topon 23256  df-bases 23291  df-cn 23572  df-cmp 23732  df-xko 23909

This theorem is used by:  cnmptkp  24026  xkofvcn  24030