Theorem yonval 18218

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 18208	. . 3 ⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
3		simpr 484	. . . . 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 2790	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (oppCat‘𝑐) = 𝑂)
7	3, 6	opeq12d 4825	. . . 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 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (HomF‘(oppCat‘𝑐)) = 𝑀)
11	7, 10	oveq12d 7378	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
12		yonval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		ovexd 7395	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝑂⟩ curryF 𝑀) ∈ V)
14	2, 11, 12, 13	fvmptd2 6950	. 2 ⊢ (𝜑 → (Yon‘𝐶) = (⟨𝐶, 𝑂⟩ curryF 𝑀))
15	1, 14	eqtrid 2784	1 ⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574  ‘cfv 6492  (class class class)co 7360  Catccat 17621  oppCatcoppc 17668   curry_F ccurf 18167  Hom_Fchof 18205  Yoncyon 18206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-yon 18208

This theorem is referenced by:  yoncl  18219  yon11  18221  yon12  18222  yon2  18223  yonpropd  18225  oppcyon  18226