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Mirrors > Home > MPE Home > Th. List > funciso | Structured version Visualization version GIF version |
Description: The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
funciso.b | β’ π΅ = (Baseβπ·) |
funciso.s | β’ πΌ = (Isoβπ·) |
funciso.t | β’ π½ = (IsoβπΈ) |
funciso.f | β’ (π β πΉ(π· Func πΈ)πΊ) |
funciso.x | β’ (π β π β π΅) |
funciso.y | β’ (π β π β π΅) |
funciso.m | β’ (π β π β (ππΌπ)) |
Ref | Expression |
---|---|
funciso | β’ (π β ((ππΊπ)βπ) β ((πΉβπ)π½(πΉβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 β’ (BaseβπΈ) = (BaseβπΈ) | |
2 | eqid 2731 | . 2 β’ (InvβπΈ) = (InvβπΈ) | |
3 | funciso.f | . . . . 5 β’ (π β πΉ(π· Func πΈ)πΊ) | |
4 | df-br 5149 | . . . . 5 β’ (πΉ(π· Func πΈ)πΊ β β¨πΉ, πΊβ© β (π· Func πΈ)) | |
5 | 3, 4 | sylib 217 | . . . 4 β’ (π β β¨πΉ, πΊβ© β (π· Func πΈ)) |
6 | funcrcl 17818 | . . . 4 β’ (β¨πΉ, πΊβ© β (π· Func πΈ) β (π· β Cat β§ πΈ β Cat)) | |
7 | 5, 6 | syl 17 | . . 3 β’ (π β (π· β Cat β§ πΈ β Cat)) |
8 | 7 | simprd 495 | . 2 β’ (π β πΈ β Cat) |
9 | funciso.b | . . . 4 β’ π΅ = (Baseβπ·) | |
10 | 9, 1, 3 | funcf1 17821 | . . 3 β’ (π β πΉ:π΅βΆ(BaseβπΈ)) |
11 | funciso.x | . . 3 β’ (π β π β π΅) | |
12 | 10, 11 | ffvelcdmd 7087 | . 2 β’ (π β (πΉβπ) β (BaseβπΈ)) |
13 | funciso.y | . . 3 β’ (π β π β π΅) | |
14 | 10, 13 | ffvelcdmd 7087 | . 2 β’ (π β (πΉβπ) β (BaseβπΈ)) |
15 | funciso.t | . 2 β’ π½ = (IsoβπΈ) | |
16 | eqid 2731 | . . 3 β’ (Invβπ·) = (Invβπ·) | |
17 | funciso.s | . . . 4 β’ πΌ = (Isoβπ·) | |
18 | 7 | simpld 494 | . . . 4 β’ (π β π· β Cat) |
19 | funciso.m | . . . 4 β’ (π β π β (ππΌπ)) | |
20 | 9, 17, 16, 18, 11, 13, 19 | invisoinvr 17743 | . . 3 β’ (π β π(π(Invβπ·)π)((π(Invβπ·)π)βπ)) |
21 | 9, 16, 2, 3, 11, 13, 20 | funcinv 17828 | . 2 β’ (π β ((ππΊπ)βπ)((πΉβπ)(InvβπΈ)(πΉβπ))((ππΊπ)β((π(Invβπ·)π)βπ))) |
22 | 1, 2, 8, 12, 14, 15, 21 | inviso1 17718 | 1 β’ (π β ((ππΊπ)βπ) β ((πΉβπ)π½(πΉβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β¨cop 4634 class class class wbr 5148 βcfv 6543 (class class class)co 7412 Basecbs 17149 Catccat 17613 Invcinv 17697 Isociso 17698 Func cfunc 17809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-map 8825 df-ixp 8895 df-cat 17617 df-cid 17618 df-sect 17699 df-inv 17700 df-iso 17701 df-func 17813 |
This theorem is referenced by: ffthiso 17885 |
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