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Mirrors > Home > MPE Home > Th. List > yonval | Structured version Visualization version GIF version |
Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.) |
Ref | Expression |
---|---|
yonval.y | ⊢ 𝑌 = (Yon‘𝐶) |
yonval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yonval.o | ⊢ 𝑂 = (oppCat‘𝐶) |
yonval.m | ⊢ 𝑀 = (HomF‘𝑂) |
Ref | Expression |
---|---|
yonval | ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yonval.y | . 2 ⊢ 𝑌 = (Yon‘𝐶) | |
2 | df-yon 18308 | . . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐)))) | |
3 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
4 | 3 | fveq2d 6911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶)) |
5 | yonval.o | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 4, 5 | eqtr4di 2793 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂) |
7 | 3, 6 | opeq12d 4886 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 〈𝑐, (oppCat‘𝑐)〉 = 〈𝐶, 𝑂〉) |
8 | 6 | fveq2d 6911 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂)) |
9 | yonval.m | . . . . 5 ⊢ 𝑀 = (HomF‘𝑂) | |
10 | 8, 9 | eqtr4di 2793 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀) |
11 | 7, 10 | oveq12d 7449 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐))) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
12 | yonval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
13 | ovexd 7466 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF 𝑀) ∈ V) | |
14 | 2, 11, 12, 13 | fvmptd2 7024 | . 2 ⊢ (𝜑 → (Yon‘𝐶) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
15 | 1, 14 | eqtrid 2787 | 1 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ‘cfv 6563 (class class class)co 7431 Catccat 17709 oppCatcoppc 17756 curryF ccurf 18267 HomFchof 18305 Yoncyon 18306 |
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-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-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 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-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-yon 18308 |
This theorem is referenced by: yoncl 18319 yon11 18321 yon12 18322 yon2 18323 yonpropd 18325 oppcyon 18326 |
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