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Mirrors > Home > MPE Home > Th. List > cidfn | Structured version Visualization version GIF version |
Description: The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
cidfn.b | ⊢ 𝐵 = (Base‘𝐶) |
cidfn.i | ⊢ 1 = (Id‘𝐶) |
Ref | Expression |
---|---|
cidfn | ⊢ (𝐶 ∈ Cat → 1 Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaex 7097 | . . 3 ⊢ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)) ∈ V | |
2 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) | |
3 | 1, 2 | fnmpti 6463 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) Fn 𝐵 |
4 | cidfn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
5 | eqid 2798 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | eqid 2798 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
7 | id 22 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
8 | cidfn.i | . . . 4 ⊢ 1 = (Id‘𝐶) | |
9 | 4, 5, 6, 7, 8 | cidfval 16939 | . . 3 ⊢ (𝐶 ∈ Cat → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓)))) |
10 | 9 | fneq1d 6416 | . 2 ⊢ (𝐶 ∈ Cat → ( 1 Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(〈𝑦, 𝑥〉(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑦)𝑔) = 𝑓))) Fn 𝐵)) |
11 | 3, 10 | mpbiri 261 | 1 ⊢ (𝐶 ∈ Cat → 1 Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 ↦ cmpt 5110 Fn wfn 6319 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Idccid 16928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-cid 16932 |
This theorem is referenced by: oppccatid 16981 fucidcl 17227 fucsect 17234 curfcl 17474 curf2ndf 17489 |
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