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Mirrors > Home > MPE Home > Th. List > cnmpt1res | Structured version Visualization version GIF version |
Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
cnmpt1res.2 | ⊢ 𝐾 = (𝐽 ↾t 𝑌) |
cnmpt1res.3 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt1res.5 | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
cnmpt1res.6 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) |
Ref | Expression |
---|---|
cnmpt1res | ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1res.5 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
2 | 1 | resmptd 6045 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴)) |
3 | cnmpt1res.6 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) | |
4 | cnmpt1res.3 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
5 | toponuni 22860 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | sseqtrd 4017 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
8 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cnrest 23233 | . . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿) ∧ 𝑌 ⊆ ∪ 𝐽) → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
10 | 3, 7, 9 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
11 | cnmpt1res.2 | . . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
12 | 11 | oveq1i 7429 | . . 3 ⊢ (𝐾 Cn 𝐿) = ((𝐽 ↾t 𝑌) Cn 𝐿) |
13 | 10, 12 | eleqtrrdi 2836 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ (𝐾 Cn 𝐿)) |
14 | 2, 13 | eqeltrrd 2826 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∪ cuni 4909 ↦ cmpt 5232 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 ↾t crest 17405 TopOnctopon 22856 Cn ccn 23172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-map 8847 df-en 8965 df-fin 8968 df-fi 9436 df-rest 17407 df-topgen 17428 df-top 22840 df-topon 22857 df-bases 22893 df-cn 23175 |
This theorem is referenced by: subgtgp 24053 symgtgp 24054 cnmptre 24892 evth2 24930 pcoass 24995 efrlim 26946 efrlimOLD 26947 ipasslem7 30718 cvxpconn 34980 cvmliftlem8 35030 |
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