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Mirrors > Home > ILE Home > Th. List > cnmptid | GIF version |
Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
Ref | Expression |
---|---|
cnmptid | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcom 1693 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | 1 | opabbii 4043 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
3 | df-id 4265 | . . . . 5 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
4 | mptv 4073 | . . . . 5 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} | |
5 | 2, 3, 4 | 3eqtr4i 2195 | . . . 4 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
6 | 5 | reseq1i 4874 | . . 3 ⊢ ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) |
7 | ssv 3159 | . . . 4 ⊢ 𝑋 ⊆ V | |
8 | resmpt 4926 | . . . 4 ⊢ (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) |
10 | 6, 9 | eqtri 2185 | . 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) |
11 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
12 | idcn 12759 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | |
13 | 11, 12 | syl 14 | . 2 ⊢ (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) |
14 | 10, 13 | eqeltrrid 2252 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 {copab 4036 ↦ cmpt 4037 I cid 4260 ↾ cres 4600 ‘cfv 5182 (class class class)co 5836 TopOnctopon 12555 Cn ccn 12732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-top 12543 df-topon 12556 df-cn 12735 |
This theorem is referenced by: imasnopn 12846 |
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