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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 16891 | . . . 4 ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐)))) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐))))) |
4 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
5 | 4 | fveq2d 6195 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶)) |
6 | yonval.o | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂) |
8 | 4, 7 | opeq12d 4410 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 〈𝑐, (oppCat‘𝑐)〉 = 〈𝐶, 𝑂〉) |
9 | 7 | fveq2d 6195 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂)) |
10 | yonval.m | . . . . 5 ⊢ 𝑀 = (HomF‘𝑂) | |
11 | 9, 10 | syl6eqr 2674 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀) |
12 | 8, 11 | oveq12d 6668 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐))) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
13 | yonval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
14 | ovexd 6680 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF 𝑀) ∈ V) | |
15 | 3, 12, 13, 14 | fvmptd 6288 | . 2 ⊢ (𝜑 → (Yon‘𝐶) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
16 | 1, 15 | syl5eq 2668 | 1 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Catccat 16325 oppCatcoppc 16371 curryF ccurf 16850 HomFchof 16888 Yoncyon 16889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-yon 16891 |
This theorem is referenced by: yoncl 16902 yon11 16904 yon12 16905 yon2 16906 yonpropd 16908 oppcyon 16909 |
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