Theorem cnmpt2res 22736

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 5595	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
5	2, 3, 4	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
6		cnmpt1res.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		cnmpt2res.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
8		txtopon 22650	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
9	6, 7, 8	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
10		toponuni 21971	. . . . 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 3957	. . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀))
13		eqid 2738	. . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀)
14	13	cnrest 22344	. . 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 583	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
16		resmpo 7372	. . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
17	2, 3, 16	syl2anc 583	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
18		topontop 21970	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	6, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
20		topontop 21970	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top)
21	7, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
22		toponmax 21983	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
23	6, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
24	23, 2	ssexd 5243	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
25		toponmax 21983	. . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀)
26	7, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
27	26, 3	ssexd 5243	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ V)
28		txrest 22690	. . . . 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 835	. . . 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 7267	. . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))
33	29, 32	eqtr4di 2797	. . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁))
34	33	oveq1d 7270	. 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 2108  Vcvv 3422   ⊆ wss 3883  ∪ cuni 4836   × cxp 5578   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ↾_t crest 17048  Topctop 21950  TopOnctopon 21967   Cn ccn 22283   ×_t ctx 22619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-map 8575  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cn 22286  df-tx 22621

This theorem is referenced by:  efmndtmd  23160  submtmd  23163  iimulcn  24007  cxpcn2  25804  cxpcn3  25806  cvxsconn  33105  cvmlift2lem6  33170  cvmlift2lem12  33176