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Mirrors > Home > ILE Home > Th. List > cnmpt2c | GIF version |
Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmpt2c.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
cnmpt2c.p | ⊢ (𝜑 → 𝑃 ∈ 𝑍) |
Ref | Expression |
---|---|
cnmpt2c | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2140 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑃 = 𝑃) | |
2 | 1 | mpompt 5863 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) |
3 | cnmpt21.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | cnmpt21.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
5 | txtopon 12431 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
6 | 3, 4, 5 | syl2anc 408 | . . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
7 | cnmpt2c.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) | |
8 | cnmpt2c.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑍) | |
9 | 6, 7, 8 | cnmptc 12451 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
10 | 2, 9 | eqeltrrid 2227 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 〈cop 3530 ↦ cmpt 3989 × cxp 4537 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 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-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-cnp 12358 df-tx 12422 |
This theorem is referenced by: cnrehmeocntop 12762 |
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