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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 5747 | . . 3 ⊢ (𝑌 × {𝑥}) = (𝑦 ∈ 𝑌 ↦ 𝑥) | |
| 2 | 1 | mpteq2i 5247 | . 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 23627 | . . 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 2846 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 {csn 4626 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 TopOnctopon 22916 Cn ccn 23232 ↑ko cxko 23569 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-map 8868 df-en 8986 df-dom 8987 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-cnp 23236 df-cmp 23395 df-xko 23571 | 
| This theorem is referenced by: (None) | 
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