Theorem cnmpt2res 12937

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 4711	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
5	2, 3, 4	syl2anc 409	. . . 4 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ (𝑋 × 𝑍))
6		cnmpt1res.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		cnmpt2res.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍))
8		txtopon 12902	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑍)) → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
9	6, 7, 8	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑍)))
10		toponuni 12653	. . . . 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 3180	. . 3 ⊢ (𝜑 → (𝑌 × 𝑊) ⊆ ∪ (𝐽 ×t 𝑀))
13		eqid 2165	. . . 4 ⊢ ∪ (𝐽 ×t 𝑀) = ∪ (𝐽 ×t 𝑀)
14	13	cnrest 12875	. . 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 409	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) ∈ (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿))
16		resmpo 5940	. . 3 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑊 ⊆ 𝑍) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
17	2, 3, 16	syl2anc 409	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ↾ (𝑌 × 𝑊)) = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴))
18		topontop 12652	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	6, 18	syl 14	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
20		topontop 12652	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑀 ∈ Top)
21	7, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
22		toponmax 12663	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
23	6, 22	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
24	23, 2	ssexd 4122	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
25		toponmax 12663	. . . . . . 7 ⊢ (𝑀 ∈ (TopOn‘𝑍) → 𝑍 ∈ 𝑀)
26	7, 25	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
27	26, 3	ssexd 4122	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ V)
28		txrest 12916	. . . . 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 1229	. . . 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 5854	. . . 4 ⊢ (𝐾 ×t 𝑁) = ((𝐽 ↾t 𝑌) ×t (𝑀 ↾t 𝑊))
33	29, 32	eqtr4di 2217	. . 3 ⊢ (𝜑 → ((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) = (𝐾 ×t 𝑁))
34	33	oveq1d 5857	. 2 ⊢ (𝜑 → (((𝐽 ×t 𝑀) ↾t (𝑌 × 𝑊)) Cn 𝐿) = ((𝐾 ×t 𝑁) Cn 𝐿))
35	15, 17, 34	3eltr3d 2249	1 ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ∪ cuni 3789   × cxp 4602   ↾ cres 4606  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844   ↾_t crest 12556  Topctop 12635  TopOnctopon 12648   Cn ccn 12825   ×_t ctx 12892

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-rest 12558  df-topgen 12577  df-top 12636  df-topon 12649  df-bases 12681  df-cn 12828  df-tx 12893

This theorem is referenced by: (None)