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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-2017.) |
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 2610 | . . 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 16247 | . 2 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑋)) |
6 | invid.i | . . 3 ⊢ 𝐼 = (Id‘𝐶) | |
7 | 1, 6, 3, 4 | invid 16270 | . 2 ⊢ (𝜑 → (𝐼‘𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼‘𝑋)) |
8 | funbrfv 6144 | . 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 63 | 1 ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼‘𝑋)) = (𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Idccid 16149 Invcinv 16228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 |
This theorem is referenced by: invisoinvl 16273 |
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