Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
6 | eqid 2818 | . . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imastopn 22256 | . . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹)) |
8 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 5 | istps 21470 | . . . . . . 7 ⊢ (𝑅 ∈ TopSp ↔ (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
10 | 4, 9 | sylib 219 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘𝑅) ∈ (TopOn‘(Base‘𝑅))) |
11 | 2 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → (TopOn‘𝑉) = (TopOn‘(Base‘𝑅))) |
12 | 10, 11 | eleqtrrd 2913 | . . . . 5 ⊢ (𝜑 → (TopOpen‘𝑅) ∈ (TopOn‘𝑉)) |
13 | qtoptopon 22240 | . . . . 5 ⊢ (((TopOpen‘𝑅) ∈ (TopOn‘𝑉) ∧ 𝐹:𝑉–onto→𝐵) → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘𝐵)) | |
14 | 12, 3, 13 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘𝐵)) |
15 | 1, 2, 3, 4 | imasbas 16773 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
16 | 15 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (TopOn‘𝐵) = (TopOn‘(Base‘𝑈))) |
17 | 14, 16 | eleqtrd 2912 | . . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) ∈ (TopOn‘(Base‘𝑈))) |
18 | 7, 17 | eqeltrd 2910 | . 2 ⊢ (𝜑 → (TopOpen‘𝑈) ∈ (TopOn‘(Base‘𝑈))) |
19 | eqid 2818 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
20 | 19, 6 | istps 21470 | . 2 ⊢ (𝑈 ∈ TopSp ↔ (TopOpen‘𝑈) ∈ (TopOn‘(Base‘𝑈))) |
21 | 18, 20 | sylibr 235 | 1 ⊢ (𝜑 → 𝑈 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TopOpenctopn 16683 qTop cqtop 16764 “s cimas 16765 TopOnctopon 21446 TopSpctps 21468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-rest 16684 df-topn 16685 df-qtop 16768 df-imas 16769 df-top 21430 df-topon 21447 df-topsp 21469 |
This theorem is referenced by: qustps 22258 xpstps 22346 |
Copyright terms: Public domain | W3C validator |