![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sectid | Structured version Visualization version GIF version |
Description: The identity is a section of itself. (Contributed by AV, 8-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 |
---|---|
sectid | ⊢ (𝜑 → (𝐼‘𝑋)(𝑋(Sect‘𝐶)𝑋)(𝐼‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2728 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | invid.i | . . 3 ⊢ 𝐼 = (Id‘𝐶) | |
4 | invid.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | invid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | eqid 2728 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | 1, 2, 3, 4, 5 | catidcl 17662 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
8 | 1, 2, 3, 4, 5, 6, 5, 7 | catlid 17663 | . 2 ⊢ (𝜑 → ((𝐼‘𝑋)(〈𝑋, 𝑋〉(comp‘𝐶)𝑋)(𝐼‘𝑋)) = (𝐼‘𝑋)) |
9 | eqid 2728 | . . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
10 | 1, 2, 6, 3, 9, 4, 5, 5, 7, 7 | issect2 17737 | . 2 ⊢ (𝜑 → ((𝐼‘𝑋)(𝑋(Sect‘𝐶)𝑋)(𝐼‘𝑋) ↔ ((𝐼‘𝑋)(〈𝑋, 𝑋〉(comp‘𝐶)𝑋)(𝐼‘𝑋)) = (𝐼‘𝑋))) |
11 | 8, 10 | mpbird 257 | 1 ⊢ (𝜑 → (𝐼‘𝑋)(𝑋(Sect‘𝐶)𝑋)(𝐼‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4635 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 Hom chom 17244 compcco 17245 Catccat 17644 Idccid 17645 Sectcsect 17727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-cat 17648 df-cid 17649 df-sect 17730 |
This theorem is referenced by: invid 17770 |
Copyright terms: Public domain | W3C validator |