Theorem yonval 18224

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 18214	. . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6888	. . . . . 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 4876	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
8	6	fveq2d 6888	. . . . 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 7422	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12		yonval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		ovexd 7439	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
14	2, 11, 12, 13	fvmptd2 6999	. 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 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629  ‘cfv 6536  (class class class)co 7404  Catccat 17615  oppCatcoppc 17662   curry_F ccurf 18173  Hom_Fchof 18211  Yoncyon 18212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-yon 18214

This theorem is referenced by:  yoncl  18225  yon11  18227  yon12  18228  yon2  18229  yonpropd  18231  oppcyon  18232