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Mirrors > Home > MPE Home > Th. List > invss | Structured version Visualization version GIF version |
Description: The inverse relation is a relation between morphisms πΉ:πβΆπ and their inverses πΊ:πβΆπ. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b | β’ π΅ = (BaseβπΆ) |
invfval.n | β’ π = (InvβπΆ) |
invfval.c | β’ (π β πΆ β Cat) |
invfval.x | β’ (π β π β π΅) |
invfval.y | β’ (π β π β π΅) |
invss.h | β’ π» = (Hom βπΆ) |
Ref | Expression |
---|---|
invss | β’ (π β (πππ) β ((ππ»π) Γ (ππ»π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
2 | invfval.n | . . . 4 β’ π = (InvβπΆ) | |
3 | invfval.c | . . . 4 β’ (π β πΆ β Cat) | |
4 | invfval.x | . . . 4 β’ (π β π β π΅) | |
5 | invfval.y | . . . 4 β’ (π β π β π΅) | |
6 | eqid 2736 | . . . 4 β’ (SectβπΆ) = (SectβπΆ) | |
7 | 1, 2, 3, 4, 5, 6 | invfval 17568 | . . 3 β’ (π β (πππ) = ((π(SectβπΆ)π) β© β‘(π(SectβπΆ)π))) |
8 | inss1 4175 | . . 3 β’ ((π(SectβπΆ)π) β© β‘(π(SectβπΆ)π)) β (π(SectβπΆ)π) | |
9 | 7, 8 | eqsstrdi 3986 | . 2 β’ (π β (πππ) β (π(SectβπΆ)π)) |
10 | invss.h | . . 3 β’ π» = (Hom βπΆ) | |
11 | eqid 2736 | . . 3 β’ (compβπΆ) = (compβπΆ) | |
12 | eqid 2736 | . . 3 β’ (IdβπΆ) = (IdβπΆ) | |
13 | 1, 10, 11, 12, 6, 3, 4, 5 | sectss 17561 | . 2 β’ (π β (π(SectβπΆ)π) β ((ππ»π) Γ (ππ»π))) |
14 | 9, 13 | sstrd 3942 | 1 β’ (π β (πππ) β ((ππ»π) Γ (ππ»π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β© cin 3897 β wss 3898 Γ cxp 5618 β‘ccnv 5619 βcfv 6479 (class class class)co 7337 Basecbs 17009 Hom chom 17070 compcco 17071 Catccat 17470 Idccid 17471 Sectcsect 17553 Invcinv 17554 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-sect 17556 df-inv 17557 |
This theorem is referenced by: invsym2 17572 invfun 17573 isohom 17585 invfuc 17789 |
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