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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 6060 | . 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 22936 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | sseqtrd 4036 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
8 | eqid 2735 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cnrest 23309 | . . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿) ∧ 𝑌 ⊆ ∪ 𝐽) → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
10 | 3, 7, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐿)) |
11 | cnmpt1res.2 | . . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
12 | 11 | oveq1i 7441 | . . 3 ⊢ (𝐾 Cn 𝐿) = ((𝐽 ↾t 𝑌) Cn 𝐿) |
13 | 10, 12 | eleqtrrdi 2850 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) ∈ (𝐾 Cn 𝐿)) |
14 | 2, 13 | eqeltrrd 2840 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 ↦ cmpt 5231 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 TopOnctopon 22932 Cn ccn 23248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-map 8867 df-en 8985 df-fin 8988 df-fi 9449 df-rest 17469 df-topgen 17490 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 |
This theorem is referenced by: subgtgp 24129 symgtgp 24130 cnmptre 24968 evth2 25006 pcoass 25071 efrlim 27027 efrlimOLD 27028 ipasslem7 30865 cvxpconn 35227 cvmliftlem8 35277 |
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