Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnmpt2nd | Structured version Visualization version GIF version |
Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
Ref | Expression |
---|---|
cnmpt2nd | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 7692 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6567 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | ssv 3939 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
5 | fnssres 6442 | . . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
6 | 3, 4, 5 | mp2an 691 | . . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
7 | dffn5 6699 | . . . 4 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))) | |
8 | 6, 7 | mpbi 233 | . . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) |
9 | fvres 6664 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧)) | |
10 | 9 | mpteq2ia 5121 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) |
11 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | op2ndd 7682 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
14 | 13 | mpompt 7245 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) |
15 | 8, 10, 14 | 3eqtri 2825 | . 2 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) |
16 | cnmpt21.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
17 | cnmpt21.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
18 | tx2cn 22215 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) | |
19 | 16, 17, 18 | syl2anc 587 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
20 | 15, 19 | eqeltrrid 2895 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ↦ cmpt 5110 × cxp 5517 ↾ cres 5521 Fn wfn 6319 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 2nd c2nd 7670 TopOnctopon 21515 Cn ccn 21829 ×t ctx 22165 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 df-tx 22167 |
This theorem is referenced by: cnmptcom 22283 xkofvcn 22289 cnmptk2 22291 txhmeo 22408 txswaphmeo 22410 ptunhmeo 22413 xkohmeo 22420 tgpsubcn 22695 istgp2 22696 oppgtmd 22702 prdstmdd 22729 dvrcn 22789 divcn 23473 cnrehmeo 23558 htpycom 23581 htpyco1 23583 htpycc 23585 reparphti 23602 pcohtpylem 23624 pcorevlem 23631 cxpcn 25334 vmcn 28482 dipcn 28503 mndpluscn 31279 cvxsconn 32603 cvmlift2lem6 32668 |
Copyright terms: Public domain | W3C validator |