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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 6663 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | idafval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2872 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶)) | |
7 | idafval.1 | . . . . . . . 8 ⊢ 1 = (Id‘𝐶) | |
8 | 6, 7 | syl6eqr 2872 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 ) |
9 | 8 | fveq1d 6665 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥)) |
10 | 9 | oteq3d 4809 | . . . . 5 ⊢ (𝑐 = 𝐶 → 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉 = 〈𝑥, 𝑥, ( 1 ‘𝑥)〉) |
11 | 5, 10 | mpteq12dv 5142 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
12 | df-ida 17307 | . . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) | |
13 | 11, 12, 4 | mptfvmpt 6982 | . . 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 2866 | 1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 〈cotp 4567 ↦ cmpt 5137 ‘cfv 6348 Basecbs 16475 Catccat 16927 Idccid 16928 Idacida 17305 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-ot 4568 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ida 17307 |
This theorem is referenced by: idaval 17310 idaf 17315 |
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