Theorem yonval 18225

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 18215	. . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = (oppCat‘𝐶))
5		yonval.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
6	4, 5	eqtr4di 2793	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
7	3, 6	opeq12d 4819	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⟨𝑐, (oppCat‘𝑐)⟩ = ⟨𝐶, 𝑂⟩)
8	6	fveq2d 6838	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = (HomF‘𝑂))
9		yonval.m	. . . . 5 ⊢ 𝑀 = (HomF‘𝑂)
10	8, 9	eqtr4di 2793	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
11	7, 10	oveq12d 7381	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12		yonval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		ovexd 7398	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
14	2, 11, 12, 13	fvmptd2 6951	. 2 ⊢ (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
15	1, 14	eqtrid 2787	1 ⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⟨cop 4568  ‘cfv 6492  (class class class)co 7363  Catccat 17628  oppCatcoppc 17675   curry_F ccurf 18174  Hom_Fchof 18212  Yoncyon 18213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-yon 18215

This theorem is referenced by:  yoncl  18226  yon11  18228  yon12  18229  yon2  18230  yonpropd  18232  oppcyon  18233