Theorem cnmpt2res 12466

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 4646	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
5	2, 3, 4	syl2anc 408	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
6		cnmpt1res.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		cnmpt2res.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
8		txtopon 12431	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
9	6, 7, 8	syl2anc 408	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
10		toponuni 12182	. . . . 5 ⊢ ((𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)) → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
11	9, 10	syl 14	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑍) = ∪ (𝐽 ×t 𝑀))
12	5, 11	sseqtrd 3135	. . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀))
13		eqid 2139	. . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀)
14	13	cnrest 12404	. . 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 408	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
16		resmpo 5869	. . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
17	2, 3, 16	syl2anc 408	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
18		topontop 12181	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	6, 18	syl 14	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
20		topontop 12181	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top)
21	7, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
22		toponmax 12192	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
23	6, 22	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
24	23, 2	ssexd 4068	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
25		toponmax 12192	. . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀)
26	7, 25	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
27	26, 3	ssexd 4068	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ V)
28		txrest 12445	. . . . 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 1217	. . . 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 5786	. . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))
33	29, 32	syl6eqr 2190	. . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁))
34	33	oveq1d 5789	. 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿))
35	15, 17, 34	3eltr3d 2222	1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  ∪ cuni 3736   × cxp 4537   ↾ cres 4541  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776   ↾_t crest 12120  Topctop 12164  TopOnctopon 12177   Cn ccn 12354   ×_t ctx 12421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-rest 12122  df-topgen 12141  df-top 12165  df-topon 12178  df-bases 12210  df-cn 12357  df-tx 12422

This theorem is referenced by: (None)