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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 | ⊢ 𝐵 = (Base‘𝐶) |
invid.i | ⊢ 𝐼 = (Id‘𝐶) |
invid.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
idinv | ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼‘𝑋)) = (𝐼‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2825 | . . 3 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | invid.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | 1, 2, 3, 4, 4 | invfun 16776 | . 2 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑋)) |
6 | invid.i | . . 3 ⊢ 𝐼 = (Id‘𝐶) | |
7 | 1, 6, 3, 4 | invid 16799 | . 2 ⊢ (𝜑 → (𝐼‘𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼‘𝑋)) |
8 | funbrfv 6480 | . 2 ⊢ (Fun (𝑋(Inv‘𝐶)𝑋) → ((𝐼‘𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼‘𝑋) → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼‘𝑋)) = (𝐼‘𝑋))) | |
9 | 5, 7, 8 | sylc 65 | 1 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼‘𝑋)) = (𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4873 Fun wfun 6117 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Catccat 16677 Idccid 16678 Invcinv 16757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-cat 16681 df-cid 16682 df-sect 16759 df-inv 16760 |
This theorem is referenced by: invisoinvl 16802 |
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