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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 6103 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | idafval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | syl6eqr 2662 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶)) | |
7 | idafval.1 | . . . . . . . 8 ⊢ 1 = (Id‘𝐶) | |
8 | 6, 7 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 ) |
9 | 8 | fveq1d 6105 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥)) |
10 | 9 | oteq3d 4354 | . . . . 5 ⊢ (𝑐 = 𝐶 → 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉 = 〈𝑥, 𝑥, ( 1 ‘𝑥)〉) |
11 | 5, 10 | mpteq12dv 4663 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
12 | df-ida 16528 | . . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) | |
13 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
14 | 4, 13 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
15 | 14 | mptex 6390 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉) ∈ V |
16 | 11, 12, 15 | fvmpt 6191 | . . 3 ⊢ (𝐶 ∈ Cat → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
17 | 2, 16 | syl 17 | . 2 ⊢ (𝜑 → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
18 | 1, 17 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cotp 4133 ↦ cmpt 4643 ‘cfv 5804 Basecbs 15695 Catccat 16148 Idccid 16149 Idacida 16526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ida 16528 |
This theorem is referenced by: idaval 16531 idaf 16536 |
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