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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 6052 | . 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) | |
2 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | idcn 23249 | . . 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 2831 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ↦ cmpt 5228 I cid 5571 ↾ cres 5676 ‘cfv 6546 (class class class)co 7416 TopOnctopon 22900 Cn ccn 23216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8849 df-top 22884 df-topon 22901 df-cn 23219 |
This theorem is referenced by: xkoinjcn 23679 txconn 23681 imasnopn 23682 imasncld 23683 imasncls 23684 pt1hmeo 23798 istgp2 24083 tmdmulg 24084 tmdlactcn 24094 clsnsg 24102 tgpt0 24111 tlmtgp 24188 nmcn 24848 expcn 24878 divccn 24879 expcnOLD 24880 divccnOLD 24881 cncfmptid 24921 cdivcncf 24929 iirevcn 24939 iihalf1cn 24941 iihalf1cnOLD 24942 iihalf2cn 24944 iihalf2cnOLD 24945 icchmeo 24953 icchmeoOLD 24954 evth2 24974 pcocn 25032 pcopt 25037 pcopt2 25038 pcoass 25039 csscld 25265 clsocv 25266 dvcnvlem 25996 resqrtcn 26774 sqrtcn 26775 efrlim 26994 efrlimOLD 26995 ipasslem7 30766 occllem 31233 hmopidmchi 32081 rmulccn 33756 cxpcncf1 34454 cvxpconn 35083 cvmlift2lem2 35145 cvmlift2lem3 35146 cvmliftphtlem 35158 knoppcnlem10 36218 cxpcncf2 45556 |
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