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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 5596 | . . 3 ⊢ (𝑌 × {𝑥}) = (𝑦 ∈ 𝑌 ↦ 𝑥) | |
2 | 1 | mpteq2i 5132 | . 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 22470 | . . 3 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑥 ∈ 𝑋 ↦ (𝑌 × {𝑥})) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) | |
6 | 3, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑌 × {𝑥})) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) |
7 | 2, 6 | eqeltrrid 2836 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 {csn 4527 ↦ cmpt 5120 × cxp 5534 ‘cfv 6358 (class class class)co 7191 TopOnctopon 21761 Cn ccn 22075 ↑ko cxko 22412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-fin 8608 df-fi 9005 df-rest 16881 df-topgen 16902 df-top 21745 df-topon 21762 df-bases 21797 df-cn 22078 df-cnp 22079 df-cmp 22238 df-xko 22414 |
This theorem is referenced by: (None) |
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