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Mirrors > Home > MPE Home > Th. List > imastps | Structured version Visualization version GIF version |
Description: The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
imastps.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imastps.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imastps.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imastps.r | ⊢ (𝜑 → 𝑅 ∈ TopSp) |
Ref | Expression |
---|---|
imastps | ⊢ (𝜑 → 𝑈 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imastps.u | . . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imastps.v | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imastps.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
4 | imastps.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ TopSp) | |
5 | eqid 2726 | . . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
6 | eqid 2726 | . . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imastopn 23712 | . . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹)) |
8 | eqid 2726 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 5 | istps 22924 | . . . . . . 7 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
10 | 4, 9 | sylib 217 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
11 | 2 | fveq2d 6897 | . . . . . 6 ⊢ (𝜑 → (TopOn‘𝑉) = (TopOn‘(Base‘𝑅))) |
12 | 10, 11 | eleqtrrd 2829 | . . . . 5 ⊢ (𝜑 → (TopOpen‘𝑅) ∈ (TopOn‘𝑉)) |
13 | qtoptopon 23696 | . . . . 5 ⊢ (((TopOpen‘𝑅) ∈ (TopOn‘𝑉) ∧ 𝐹:𝑉–onto→𝐵) → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘𝐵)) | |
14 | 12, 3, 13 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘𝐵)) |
15 | 1, 2, 3, 4 | imasbas 17522 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
16 | 15 | fveq2d 6897 | . . . 4 ⊢ (𝜑 → (TopOn‘𝐵) = (TopOn‘(Base‘𝑈))) |
17 | 14, 16 | eleqtrd 2828 | . . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘(Base‘𝑈))) |
18 | 7, 17 | eqeltrd 2826 | . 2 ⊢ (𝜑 → (TopOpen‘𝑈) ∈ (TopOn‘(Base‘𝑈))) |
19 | eqid 2726 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
20 | 19, 6 | istps 22924 | . 2 ⊢ (𝑈 ∈ TopSp ↔ (TopOpen‘𝑈) ∈ (TopOn‘(Base‘𝑈))) |
21 | 18, 20 | sylibr 233 | 1 ⊢ (𝜑 → 𝑈 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 –onto→wfo 6544 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 TopOpenctopn 17431 qTop cqtop 17513 “s cimas 17514 TopOnctopon 22900 TopSpctps 22922 |
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-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 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-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-struct 17144 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-rest 17432 df-topn 17433 df-qtop 17517 df-imas 17518 df-top 22884 df-topon 22901 df-topsp 22923 |
This theorem is referenced by: qustps 23714 xpstps 23802 |
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