![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > thincid | Structured version Visualization version GIF version |
Description: In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincid.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincid.b | ⊢ 𝐵 = (Base‘𝐶) |
thincid.h | ⊢ 𝐻 = (Hom ‘𝐶) |
thincid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincid.i | ⊢ 1 = (Id‘𝐶) |
thincid.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋)) |
Ref | Expression |
---|---|
thincid | ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincid.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | thincid.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋)) | |
3 | thincid.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
4 | thincid.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | thincid.i | . . 3 ⊢ 1 = (Id‘𝐶) | |
6 | thincid.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
7 | 6 | thinccd 46940 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
8 | 3, 4, 5, 7, 1 | catidcl 17522 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
9 | 1, 1, 2, 8, 3, 4, 6 | thincmo2 46943 | 1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 Hom chom 17104 Idccid 17505 ThinCatcthinc 46934 |
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-pr 5383 |
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-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 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-cat 17508 df-cid 17509 df-thinc 46935 |
This theorem is referenced by: functhinclem4 46959 thincsect 46972 |
Copyright terms: Public domain | W3C validator |