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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 6069 | . 2 ⊢ ( I ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝑥) | |
| 2 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 3 | idcn 23265 | . . 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 2846 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ↦ cmpt 5225 I cid 5577 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 TopOnctopon 22916 Cn ccn 23232 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-top 22900 df-topon 22917 df-cn 23235 |
| This theorem is referenced by: xkoinjcn 23695 txconn 23697 imasnopn 23698 imasncld 23699 imasncls 23700 pt1hmeo 23814 istgp2 24099 tmdmulg 24100 tmdlactcn 24110 clsnsg 24118 tgpt0 24127 tlmtgp 24204 nmcn 24866 expcn 24896 divccn 24897 expcnOLD 24898 divccnOLD 24899 cncfmptid 24939 cdivcncf 24947 iirevcn 24957 iihalf1cn 24959 iihalf1cnOLD 24960 iihalf2cn 24962 iihalf2cnOLD 24963 icchmeo 24971 icchmeoOLD 24972 evth2 24992 pcocn 25050 pcopt 25055 pcopt2 25056 pcoass 25057 csscld 25283 clsocv 25284 dvcnvlem 26014 resqrtcn 26792 sqrtcn 26793 efrlim 27012 efrlimOLD 27013 ipasslem7 30855 occllem 31322 hmopidmchi 32170 rmulccn 33927 cxpcncf1 34610 cvxpconn 35247 cvmlift2lem2 35309 cvmlift2lem3 35310 cvmliftphtlem 35322 knoppcnlem10 36503 cxpcncf2 45914 |
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