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| 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 18296 | . . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐)))) | |
| 3 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
| 4 | 3 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶)) | 
| 5 | yonval.o | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂) | 
| 7 | 3, 6 | opeq12d 4881 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 〈𝑐, (oppCat‘𝑐)〉 = 〈𝐶, 𝑂〉) | 
| 8 | 6 | fveq2d 6910 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂)) | 
| 9 | yonval.m | . . . . 5 ⊢ 𝑀 = (HomF‘𝑂) | |
| 10 | 8, 9 | eqtr4di 2795 | . . . 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 2789 | 1 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 (class class class)co 7431 Catccat 17707 oppCatcoppc 17754 curryF ccurf 18255 HomFchof 18293 Yoncyon 18294 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-yon 18296 | 
| This theorem is referenced by: yoncl 18307 yon11 18309 yon12 18310 yon2 18311 yonpropd 18313 oppcyon 18314 | 
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