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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 6907 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
4 | idafval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
6 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶)) | |
7 | idafval.1 | . . . . . . . 8 ⊢ 1 = (Id‘𝐶) | |
8 | 6, 7 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 ) |
9 | 8 | fveq1d 6909 | . . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥)) |
10 | 9 | oteq3d 4892 | . . . . 5 ⊢ (𝑐 = 𝐶 → 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉 = 〈𝑥, 𝑥, ( 1 ‘𝑥)〉) |
11 | 5, 10 | mpteq12dv 5239 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉) = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
12 | df-ida 18109 | . . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) | |
13 | 11, 12, 4 | mptfvmpt 7248 | . . 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 | eqtrid 2787 | 1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 〈cotp 4639 ↦ cmpt 5231 ‘cfv 6563 Basecbs 17245 Catccat 17709 Idccid 17710 Idacida 18107 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ida 18109 |
This theorem is referenced by: idaval 18112 idaf 18117 |
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