Theorem cnmpt2res 24051

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 5673	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
5	2, 3, 4	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
6		cnmpt1res.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		cnmpt2res.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
8		txtopon 23965	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
9	6, 7, 8	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
10		toponuni 23287	. . . . 5 ⊢ ((𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)) → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
12	5, 11	sseqtrd 3949	. . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀))
13		eqid 2763	. . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀)
14	13	cnrest 23658	. . 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 596	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
16		resmpo 7571	. . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
17	2, 3, 16	syl2anc 596	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
18		topontop 23286	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	6, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
20		topontop 23286	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top)
21	7, 20	syl 18	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
22		toponmax 23299	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
23	6, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
24	23, 2	ssexd 5293	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
25		toponmax 23299	. . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀)
26	7, 25	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
27	26, 3	ssexd 5293	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ V)
28		txrest 24005	. . . . 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 852	. . . 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 7463	. . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))
33	29, 32	eqtr4di 2816	. . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁))
34	33	oveq1d 7466	. 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿))
35	15, 17, 34	3eltr3d 2878	1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∪ cuni 4872   × cxp 5656   ↾ cres 5660  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ↾_t crest 17641  Topctop 23266  TopOnctopon 23283   Cn ccn 23597   ×_t ctx 23934

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-map 8887  df-en 9012  df-fin 9015  df-fi 9440  df-rest 17643  df-topgen 17664  df-top 23267  df-topon 23284  df-bases 23319  df-cn 23600  df-tx 23936

This theorem is used by:  efmndtmd  24475  submtmd  24478  iimulcn  25314  cxpcn2  27131  cxpcn3  27133  cvxsconn  36216  cvmlift2lem6  36281  cvmlift2lem12  36287