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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 | mptresid 5903 | . 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) | |
2 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | idcn 22108 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) |
5 | 1, 4 | eqeltrrid 2836 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ↦ cmpt 5120 I cid 5439 ↾ cres 5538 ‘cfv 6358 (class class class)co 7191 TopOnctopon 21761 Cn ccn 22075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-top 21745 df-topon 21762 df-cn 22078 |
This theorem is referenced by: xkoinjcn 22538 txconn 22540 imasnopn 22541 imasncld 22542 imasncls 22543 pt1hmeo 22657 istgp2 22942 tmdmulg 22943 tmdlactcn 22953 clsnsg 22961 tgpt0 22970 tlmtgp 23047 nmcn 23695 expcn 23723 divccn 23724 cncfmptid 23764 cdivcncf 23772 iirevcn 23781 iihalf1cn 23783 iihalf2cn 23785 icchmeo 23792 evth2 23811 pcocn 23868 pcopt 23873 pcopt2 23874 pcoass 23875 csscld 24100 clsocv 24101 dvcnvlem 24827 resqrtcn 25589 sqrtcn 25590 efrlim 25806 ipasslem7 28871 occllem 29338 hmopidmchi 30186 rmulccn 31546 cxpcncf1 32241 cvxpconn 32871 cvmlift2lem2 32933 cvmlift2lem3 32934 cvmliftphtlem 32946 knoppcnlem10 34368 cxpcncf2 43058 |
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