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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 17503 | . . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐)))) | |
3 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
4 | 3 | fveq2d 6676 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶)) |
5 | yonval.o | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 4, 5 | syl6eqr 2876 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂) |
7 | 3, 6 | opeq12d 4813 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 〈𝑐, (oppCat‘𝑐)〉 = 〈𝐶, 𝑂〉) |
8 | 6 | fveq2d 6676 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂)) |
9 | yonval.m | . . . . 5 ⊢ 𝑀 = (HomF‘𝑂) | |
10 | 8, 9 | syl6eqr 2876 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀) |
11 | 7, 10 | oveq12d 7176 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐))) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
12 | yonval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
13 | ovexd 7193 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF 𝑀) ∈ V) | |
14 | 2, 11, 12, 13 | fvmptd2 6778 | . 2 ⊢ (𝜑 → (Yon‘𝐶) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
15 | 1, 14 | syl5eq 2870 | 1 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ‘cfv 6357 (class class class)co 7158 Catccat 16937 oppCatcoppc 16983 curryF ccurf 17462 HomFchof 17500 Yoncyon 17501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-yon 17503 |
This theorem is referenced by: yoncl 17514 yon11 17516 yon12 17517 yon2 17518 yonpropd 17520 oppcyon 17521 |
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