Theorem yonval 18222

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 18212	. . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
5		yonval.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
6	4, 5	eqtr4di 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
7	3, 6	opeq12d 4845	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
8	6	fveq2d 6862	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂))
9		yonval.m	. . . . 5 ⊢ 𝑀 = (HomF‘𝑂)
10	8, 9	eqtr4di 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
11	7, 10	oveq12d 7405	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12		yonval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		ovexd 7422	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
14	2, 11, 12, 13	fvmptd2 6976	. 2 ⊢ (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
15	1, 14	eqtrid 2776	1 ⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595  ‘cfv 6511  (class class class)co 7387  Catccat 17625  oppCatcoppc 17672   curry_F ccurf 18171  Hom_Fchof 18209  Yoncyon 18210

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-yon 18212

This theorem is referenced by:  yoncl  18223  yon11  18225  yon12  18226  yon2  18227  yonpropd  18229  oppcyon  18230