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Mirrors > Home > MPE Home > Th. List > cnmptid | Structured version Visualization version 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 1932 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | 1 | opabbii 4649 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
3 | dfid3 4954 | . . . . 5 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
4 | mptv 4679 | . . . . 5 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} | |
5 | 2, 3, 4 | 3eqtr4i 2642 | . . . 4 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
6 | 5 | reseq1i 5313 | . . 3 ⊢ ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) |
7 | ssv 3588 | . . . 4 ⊢ 𝑋 ⊆ V | |
8 | resmpt 5369 | . . . 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 2632 | . 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) |
11 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
12 | idcn 20871 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) |
14 | 10, 13 | syl5eqelr 2693 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {copab 4642 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 TopOnctopon 20518 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-top 20521 df-topon 20523 df-cn 20841 |
This theorem is referenced by: xkoinjcn 21300 txcon 21302 imasnopn 21303 imasncld 21304 imasncls 21305 pt1hmeo 21419 istgp2 21705 tmdmulg 21706 tmdlactcn 21716 clsnsg 21723 tgpt0 21732 tlmtgp 21809 nmcn 22455 expcn 22483 divccn 22484 cncfmptid 22523 cdivcncf 22528 iirevcn 22537 iihalf1cn 22539 iihalf2cn 22541 icchmeo 22548 evth2 22567 pcocn 22625 pcopt 22630 pcopt2 22631 pcoass 22632 csscld 22856 clsocv 22857 dvcnvlem 23543 resqrtcn 24290 sqrtcn 24291 efrlim 24496 ipasslem7 27075 occllem 27546 hmopidmchi 28394 rmulccn 29302 cvxpcon 30478 cvmlift2lem2 30540 cvmlift2lem3 30541 cvmliftphtlem 30553 knoppcnlem10 31662 cxpcncf2 38786 |
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