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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 7188 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6117 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | ssv 3625 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
5 | fnssres 6004 | . . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
6 | 3, 4, 5 | mp2an 708 | . . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
7 | dffn5 6241 | . . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))) | |
8 | 6, 7 | mpbi 220 | . . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) |
9 | fvres 6207 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧)) | |
10 | 9 | mpteq2ia 4740 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) |
11 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | op1std 7178 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
14 | 13 | mpt2mpt 6752 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) |
15 | 8, 10, 14 | 3eqtri 2648 | . 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 21412 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) | |
19 | 16, 17, 18 | syl2anc 693 | . 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
20 | 15, 19 | syl5eqelr 2706 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 Fn wfn 5883 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 TopOnctopon 20715 Cn ccn 21028 ×t ctx 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-tx 21365 |
This theorem is referenced by: cnmptcom 21481 xkofvcn 21487 cnmptk2 21489 txhmeo 21606 txswaphmeo 21608 ptunhmeo 21611 xkohmeo 21618 tgpsubcn 21894 istgp2 21895 oppgtmd 21901 prdstmdd 21927 dvrcn 21987 divcn 22671 cnrehmeo 22752 htpycom 22775 htpyid 22776 htpyco1 22777 htpycc 22779 reparphti 22797 pcocn 22817 pcohtpylem 22819 pcopt 22822 pcopt2 22823 pcoass 22824 pcorevlem 22826 cxpcn 24486 vmcn 27554 dipcn 27575 mndpluscn 29972 cvxsconn 31225 cvmlift2lem12 31296 |
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