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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 8034 | . . . . . 6 ⊢ 1st :V–onto→V | |
| 2 | fofn 6822 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
| 3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
| 4 | ssv 4008 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
| 5 | fnssres 6691 | . . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
| 6 | 3, 4, 5 | mp2an 692 | . . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
| 7 | dffn5 6967 | . . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))) | |
| 8 | 6, 7 | mpbi 230 | . . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) |
| 9 | fvres 6925 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧)) | |
| 10 | 9 | mpteq2ia 5245 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) |
| 11 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 12 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 13 | 11, 12 | op1std 8024 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
| 14 | 13 | mpompt 7547 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) |
| 15 | 8, 10, 14 | 3eqtri 2769 | . 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 23617 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) | |
| 19 | 16, 17, 18 | syl2anc 584 | . 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
| 20 | 15, 19 | eqeltrrid 2846 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 |
| 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-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-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-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-tx 23570 |
| This theorem is referenced by: cnmptcom 23686 xkofvcn 23692 cnmptk2 23694 txhmeo 23811 txswaphmeo 23813 ptunhmeo 23816 xkohmeo 23823 tgpsubcn 24098 istgp2 24099 oppgtmd 24105 prdstmdd 24132 dvrcn 24192 divcnOLD 24890 divcn 24892 cnrehmeo 24984 cnrehmeoOLD 24985 htpycom 25008 htpyid 25009 htpyco1 25010 htpycc 25012 reparphti 25029 reparphtiOLD 25030 pcocn 25050 pcohtpylem 25052 pcopt 25055 pcopt2 25056 pcoass 25057 pcorevlem 25059 cxpcn 26787 cxpcnOLD 26788 vmcn 30718 dipcn 30739 mndpluscn 33925 cvxsconn 35248 cvmlift2lem12 35319 |
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