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Mirrors > Home > MPE Home > Th. List > idafval | Structured version Visualization version GIF version |
Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
idafval.i | ⊢ 𝐼 = (Ida‘𝐶) |
idafval.b | ⊢ 𝐵 = (Base‘𝐶) |
idafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idafval.1 | ⊢ 1 = (Id‘𝐶) |
Ref | Expression |
---|---|
idafval | ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idafval.i | . 2 ⊢ 𝐼 = (Ida‘𝐶) | |
2 | idafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | fveq2 6664 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | idafval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶)) | |
7 | idafval.1 | . . . . . . . 8 ⊢ 1 = (Id‘𝐶) | |
8 | 6, 7 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 ) |
9 | 8 | fveq1d 6666 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥)) |
10 | 9 | oteq3d 4810 | . . . . 5 ⊢ (𝑐 = 𝐶 → 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉 = 〈𝑥, 𝑥, ( 1 ‘𝑥)〉) |
11 | 5, 10 | mpteq12dv 5143 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
12 | df-ida 17309 | . . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) | |
13 | 11, 12, 4 | mptfvmpt 6984 | . . 3 ⊢ (𝐶 ∈ Cat → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝜑 → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
15 | 1, 14 | syl5eq 2868 | 1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cotp 4568 ↦ cmpt 5138 ‘cfv 6349 Basecbs 16477 Catccat 16929 Idccid 16930 Idacida 17307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-ot 4569 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ida 17309 |
This theorem is referenced by: idaval 17312 idaf 17317 |
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