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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 7712 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6594 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | ssv 3993 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
5 | fnssres 6472 | . . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
6 | 3, 4, 5 | mp2an 690 | . . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
7 | dffn5 6726 | . . . 4 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))) | |
8 | 6, 7 | mpbi 232 | . . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) |
9 | fvres 6691 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧)) | |
10 | 9 | mpteq2ia 5159 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) |
11 | vex 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | op2ndd 7702 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
14 | 13 | mpompt 7268 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) |
15 | 8, 10, 14 | 3eqtri 2850 | . 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 22220 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) | |
19 | 16, 17, 18 | syl2anc 586 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
20 | 15, 19 | eqeltrrid 2920 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ↦ cmpt 5148 × cxp 5555 ↾ cres 5559 Fn wfn 6352 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 2nd c2nd 7690 TopOnctopon 21520 Cn ccn 21834 ×t ctx 22170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cn 21837 df-tx 22172 |
This theorem is referenced by: cnmptcom 22288 xkofvcn 22294 cnmptk2 22296 txhmeo 22413 txswaphmeo 22415 ptunhmeo 22418 xkohmeo 22425 tgpsubcn 22700 istgp2 22701 oppgtmd 22707 prdstmdd 22734 dvrcn 22794 divcn 23478 cnrehmeo 23559 htpycom 23582 htpyco1 23584 htpycc 23586 reparphti 23603 pcohtpylem 23625 pcorevlem 23632 cxpcn 25328 vmcn 28478 dipcn 28499 mndpluscn 31171 cvxsconn 32492 cvmlift2lem6 32557 |
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