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Mirrors > Home > MPE Home > Th. List > cnmptkc | Structured version Visualization version GIF version |
Description: The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptk1.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmptk1.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
Ref | Expression |
---|---|
cnmptkc | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5692 | . . 3 ⊢ (𝑌 × {𝑥}) = (𝑦 ∈ 𝑌 ↦ 𝑥) | |
2 | 1 | mpteq2i 5208 | . 2 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑌 × {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) |
3 | cnmptk1.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
4 | cnmptk1.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
5 | xkoccn 22954 | . . 3 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑥 ∈ 𝑋 ↦ (𝑌 × {𝑥})) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) | |
6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑌 × {𝑥})) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) |
7 | 2, 6 | eqeltrrid 2843 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 {csn 4584 ↦ cmpt 5186 × cxp 5629 ‘cfv 6493 (class class class)co 7353 TopOnctopon 22243 Cn ccn 22559 ↑ko cxko 22896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-fin 8883 df-fi 9343 df-rest 17296 df-topgen 17317 df-top 22227 df-topon 22244 df-bases 22280 df-cn 22562 df-cnp 22563 df-cmp 22722 df-xko 22898 |
This theorem is referenced by: (None) |
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