Theorem diagval 18296

Description: Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
diagval.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
diagval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diagval.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
diagval	⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

Proof of Theorem diagval

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diagval.l	. 2 ⊢ 𝐿 = (𝐶Δfunc𝐷)
2		df-diag 18272	. . . 4 ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)))
3	2	a1i 11	. . 3 ⊢ (𝜑 → Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑))))
4		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	opeq12d 4885	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
7	4, 5	oveq12d 7448	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷))
8	6, 7	oveq12d 7448	. . 3 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
9		diagval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		diagval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
11		ovexd 7465	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) ∈ V)
12	3, 8, 9, 10, 11	ovmpod 7584	. 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
13	1, 12	eqtrid 2786	1 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  Vcvv 3477  ⟨cop 4636  (class class class)co 7430   ∈ cmpo 7432  Catccat 17708   1^st_F c1stf 18224   curry_F ccurf 18266  Δ_funccdiag 18268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-diag 18272

This theorem is referenced by:  diagcl  18297  diag11  18299  diag12  18300  diag2  18301