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Mirrors > Home > MPE Home > Th. List > inviso2 | Structured version Visualization version GIF version |
Description: If πΊ is an inverse to πΉ, then πΊ is an isomorphism. (Contributed by Mario Carneiro, 3-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΅) |
isoval.n | β’ πΌ = (IsoβπΆ) |
inviso1.1 | β’ (π β πΉ(πππ)πΊ) |
Ref | Expression |
---|---|
inviso2 | β’ (π β πΊ β (ππΌπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . 2 β’ π΅ = (BaseβπΆ) | |
2 | invfval.n | . 2 β’ π = (InvβπΆ) | |
3 | invfval.c | . 2 β’ (π β πΆ β Cat) | |
4 | invfval.y | . 2 β’ (π β π β π΅) | |
5 | invfval.x | . 2 β’ (π β π β π΅) | |
6 | isoval.n | . 2 β’ πΌ = (IsoβπΆ) | |
7 | inviso1.1 | . . 3 β’ (π β πΉ(πππ)πΊ) | |
8 | 1, 2, 3, 5, 4 | invsym 17752 | . . 3 β’ (π β (πΉ(πππ)πΊ β πΊ(πππ)πΉ)) |
9 | 7, 8 | mpbid 231 | . 2 β’ (π β πΊ(πππ)πΉ) |
10 | 1, 2, 3, 4, 5, 6, 9 | inviso1 17756 | 1 β’ (π β πΊ β (ππΌπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17187 Catccat 17651 Invcinv 17735 Isociso 17736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-cat 17655 df-cid 17656 df-sect 17737 df-inv 17738 df-iso 17739 |
This theorem is referenced by: yonffthlem 18281 |
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