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Mirrors > Home > MPE Home > Th. List > cnmpt1st | Structured version Visualization version GIF version |
Description: The projection onto the first 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 |
---|---|
cnmpt1st | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 7709 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6592 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | ssv 3991 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
5 | fnssres 6470 | . . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
6 | 3, 4, 5 | mp2an 690 | . . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
7 | dffn5 6724 | . . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))) | |
8 | 6, 7 | mpbi 232 | . . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) |
9 | fvres 6689 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧)) | |
10 | 9 | mpteq2ia 5157 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) |
11 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | op1std 7699 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
14 | 13 | mpompt 7266 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) |
15 | 8, 10, 14 | 3eqtri 2848 | . 2 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) |
16 | cnmpt21.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
17 | cnmpt21.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
18 | tx1cn 22217 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) | |
19 | 16, 17, 18 | syl2anc 586 | . 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
20 | 15, 19 | eqeltrrid 2918 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ↦ cmpt 5146 × cxp 5553 ↾ cres 5557 Fn wfn 6350 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-tx 22170 |
This theorem is referenced by: cnmptcom 22286 xkofvcn 22292 cnmptk2 22294 txhmeo 22411 txswaphmeo 22413 ptunhmeo 22416 xkohmeo 22423 tgpsubcn 22698 istgp2 22699 oppgtmd 22705 prdstmdd 22732 dvrcn 22792 divcn 23476 cnrehmeo 23557 htpycom 23580 htpyid 23581 htpyco1 23582 htpycc 23584 reparphti 23601 pcocn 23621 pcohtpylem 23623 pcopt 23626 pcopt2 23627 pcoass 23628 pcorevlem 23630 cxpcn 25326 vmcn 28476 dipcn 28497 mndpluscn 31169 cvxsconn 32490 cvmlift2lem12 32561 |
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