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Mirrors > Home > MPE Home > Th. List > inviso1 | Structured version Visualization version GIF version |
Description: If πΊ is an inverse to πΉ, then πΉ 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 |
---|---|
inviso1 | β’ (π β πΉ β (ππΌπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 β’ π΅ = (BaseβπΆ) | |
2 | invfval.n | . . . . 5 β’ π = (InvβπΆ) | |
3 | invfval.c | . . . . 5 β’ (π β πΆ β Cat) | |
4 | invfval.x | . . . . 5 β’ (π β π β π΅) | |
5 | invfval.y | . . . . 5 β’ (π β π β π΅) | |
6 | 1, 2, 3, 4, 5 | invfun 17582 | . . . 4 β’ (π β Fun (πππ)) |
7 | funrel 6514 | . . . 4 β’ (Fun (πππ) β Rel (πππ)) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π β Rel (πππ)) |
9 | inviso1.1 | . . 3 β’ (π β πΉ(πππ)πΊ) | |
10 | releldm 5896 | . . 3 β’ ((Rel (πππ) β§ πΉ(πππ)πΊ) β πΉ β dom (πππ)) | |
11 | 8, 9, 10 | syl2anc 585 | . 2 β’ (π β πΉ β dom (πππ)) |
12 | isoval.n | . . 3 β’ πΌ = (IsoβπΆ) | |
13 | 1, 2, 3, 4, 5, 12 | isoval 17583 | . 2 β’ (π β (ππΌπ) = dom (πππ)) |
14 | 11, 13 | eleqtrrd 2842 | 1 β’ (π β πΉ β (ππΌπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5104 dom cdm 5631 Rel wrel 5636 Fun wfun 6486 βcfv 6492 (class class class)co 7350 Basecbs 17018 Catccat 17479 Invcinv 17563 Isociso 17564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-1st 7912 df-2nd 7913 df-cat 17483 df-cid 17484 df-sect 17565 df-inv 17566 df-iso 17567 |
This theorem is referenced by: inviso2 17585 isoco 17595 idiso 17606 funciso 17695 ffthiso 17751 fuciso 17799 initoeu1 17832 termoeu1 17839 catciso 17932 yoneda 18107 |
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