Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 16714 | . . . 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 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
5 | 4 | fveq2d 6107 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶)) |
6 | yonval.o | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂) |
8 | 4, 7 | opeq12d 4348 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 〈𝑐, (oppCat‘𝑐)〉 = 〈𝐶, 𝑂〉) |
9 | 7 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂)) |
10 | yonval.m | . . . . 5 ⊢ 𝑀 = (HomF‘𝑂) | |
11 | 9, 10 | syl6eqr 2662 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀) |
12 | 8, 11 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐))) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
13 | yonval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
14 | ovex 6577 | . . . 4 ⊢ (〈𝐶, 𝑂〉 curryF 𝑀) ∈ V | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝑂〉 curryF 𝑀) ∈ V) |
16 | 3, 12, 13, 15 | fvmptd 6197 | . 2 ⊢ (𝜑 → (Yon‘𝐶) = (〈𝐶, 𝑂〉 curryF 𝑀)) |
17 | 1, 16 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝑌 = (〈𝐶, 𝑂〉 curryF 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Catccat 16148 oppCatcoppc 16194 curryF ccurf 16673 HomFchof 16711 Yoncyon 16712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-yon 16714 |
This theorem is referenced by: yoncl 16725 yon11 16727 yon12 16728 yon2 16729 yonpropd 16731 oppcyon 16732 |
Copyright terms: Public domain | W3C validator |