Theorem yonval 18164

Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
yonval.y	⊢ 𝑌 = (Yon‘𝐶)
yonval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yonval.o	⊢ 𝑂 = (oppCat‘𝐶)
yonval.m	⊢ 𝑀 = (HomF‘𝑂)

Assertion

Ref	Expression
yonval	⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Proof of Theorem yonval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		yonval.y	. 2 ⊢ 𝑌 = (Yon‘𝐶)
2		df-yon 18154	. . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
5		yonval.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
6	4, 5	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
7	3, 6	opeq12d 4833	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
8	6	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂))
9		yonval.m	. . . . 5 ⊢ 𝑀 = (HomF‘𝑂)
10	8, 9	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
11	7, 10	oveq12d 7364	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12		yonval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		ovexd 7381	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
14	2, 11, 12, 13	fvmptd2 6937	. 2 ⊢ (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
15	1, 14	eqtrid 2778	1 ⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4582  ‘cfv 6481  (class class class)co 7346  Catccat 17567  oppCatcoppc 17614   curry_F ccurf 18113  Hom_Fchof 18151  Yoncyon 18152

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-yon 18154

This theorem is referenced by:  yoncl  18165  yon11  18167  yon12  18168  yon2  18169  yonpropd  18171  oppcyon  18172