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Mirrors > Home > MPE Home > Th. List > cnmpt12f | Structured version Visualization version GIF version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt11.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) |
cnmpt1t.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) |
cnmpt12f.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) |
Ref | Expression |
---|---|
cnmpt12f | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | mpteq2i 5122 | . 2 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) |
3 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | cnmpt11.a | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) | |
5 | cnmpt1t.b | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) | |
6 | 3, 4, 5 | cnmpt1t 22270 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ (𝐽 Cn (𝐾 ×t 𝐿))) |
7 | cnmpt12f.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) | |
8 | 3, 6, 7 | cnmpt11f 22269 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) ∈ (𝐽 Cn 𝑀)) |
9 | 2, 8 | eqeltrid 2894 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 〈cop 4531 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 TopOnctopon 21515 Cn ccn 21829 ×t ctx 22165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 df-tx 22167 |
This theorem is referenced by: cnmpt12 22272 cnmpt1plusg 22692 istgp2 22696 clsnsg 22715 tgpt0 22724 cnmpt1vsca 22799 cnmpt1ds 23447 fsumcn 23475 expcn 23477 divccn 23478 cncfmpt2f 23520 cdivcncf 23526 iirevcn 23535 iihalf1cn 23537 iihalf2cn 23539 icchmeo 23546 evth 23564 evth2 23565 pcoass 23629 cnmpt1ip 23851 dvcnvlem 24579 plycn 24858 psercn2 25018 atansopn 25518 efrlim 25555 ipasslem7 28619 occllem 29086 hmopidmchi 29934 cvxpconn 32602 cvmlift2lem2 32664 cvmlift2lem3 32665 cvmliftphtlem 32677 sinccvglem 33028 knoppcnlem10 33954 broucube 35091 areacirclem2 35146 fprodcnlem 42241 |
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