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Mirrors > Home > MPE Home > Th. List > idinv | Structured version Visualization version GIF version |
Description: The inverse of the identity is the identity. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 9-Apr-2020.) |
Ref | Expression |
---|---|
invid.b | β’ π΅ = (BaseβπΆ) |
invid.i | β’ πΌ = (IdβπΆ) |
invid.c | β’ (π β πΆ β Cat) |
invid.x | β’ (π β π β π΅) |
Ref | Expression |
---|---|
idinv | β’ (π β ((π(InvβπΆ)π)β(πΌβπ)) = (πΌβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | . . 3 β’ π΅ = (BaseβπΆ) | |
2 | eqid 2738 | . . 3 β’ (InvβπΆ) = (InvβπΆ) | |
3 | invid.c | . . 3 β’ (π β πΆ β Cat) | |
4 | invid.x | . . 3 β’ (π β π β π΅) | |
5 | 1, 2, 3, 4, 4 | invfun 17582 | . 2 β’ (π β Fun (π(InvβπΆ)π)) |
6 | invid.i | . . 3 β’ πΌ = (IdβπΆ) | |
7 | 1, 6, 3, 4 | invid 17605 | . 2 β’ (π β (πΌβπ)(π(InvβπΆ)π)(πΌβπ)) |
8 | funbrfv 6889 | . 2 β’ (Fun (π(InvβπΆ)π) β ((πΌβπ)(π(InvβπΆ)π)(πΌβπ) β ((π(InvβπΆ)π)β(πΌβπ)) = (πΌβπ))) | |
9 | 5, 7, 8 | sylc 65 | 1 β’ (π β ((π(InvβπΆ)π)β(πΌβπ)) = (πΌβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5104 Fun wfun 6486 βcfv 6492 (class class class)co 7350 Basecbs 17018 Catccat 17479 Idccid 17480 Invcinv 17563 |
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 |
This theorem is referenced by: invisoinvl 17608 |
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