Theorem cnmpt2res 22828

Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)

Hypotheses

Ref	Expression
cnmpt1res.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)
cnmpt1res.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt1res.5	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
cnmpt2res.7	⊢ 𝑁 = (𝑀 ↾t 𝑊)
cnmpt2res.8	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
cnmpt2res.9	⊢ (𝜑 → 𝑊 ⊆ 𝑍)
cnmpt2res.10	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿))

Assertion

Ref	Expression
cnmpt2res	⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

Distinct variable groups:   𝑥,𝑦,𝑊   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem cnmpt2res

Step	Hyp	Ref	Expression
1		cnmpt2res.10	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿))
2		cnmpt1res.5	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
3		cnmpt2res.9	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
4		xpss12 5604	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
6		cnmpt1res.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		cnmpt2res.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
8		txtopon 22742	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
9	6, 7, 8	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
10		toponuni 22063	. . . . 5 ⊢ ((𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)) → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
12	5, 11	sseqtrd 3961	. . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀))
13		eqid 2738	. . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀)
14	13	cnrest 22436	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿) ∧ (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
15	1, 12, 14	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
16		resmpo 7394	. . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
17	2, 3, 16	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
18		topontop 22062	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	6, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
20		topontop 22062	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top)
21	7, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
22		toponmax 22075	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
23	6, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
24	23, 2	ssexd 5248	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
25		toponmax 22075	. . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀)
26	7, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
27	26, 3	ssexd 5248	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ V)
28		txrest 22782	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑀 ∈ Top) ∧ (𝑌 ∈ V ∧ 𝑊 ∈ V)) → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)))
29	19, 21, 24, 27, 28	syl22anc 836	. . . 4 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊)))
30		cnmpt1res.2	. . . . 5 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
31		cnmpt2res.7	. . . . 5 ⊢ 𝑁 = (𝑀 ↾t 𝑊)
32	30, 31	oveq12i 7287	. . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))
33	29, 32	eqtr4di 2796	. . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁))
34	33	oveq1d 7290	. 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿))
35	15, 17, 34	3eltr3d 2853	1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   × cxp 5587   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↾_t crest 17131  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8617  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-tx 22713

This theorem is referenced by:  efmndtmd  23252  submtmd  23255  iimulcn  24101  cxpcn2  25899  cxpcn3  25901  cvxsconn  33205  cvmlift2lem6  33270  cvmlift2lem12  33276