Theorem diagval 18167

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 18143	. . . 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 4838	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
7	4, 5	oveq12d 7378	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷))
8	6, 7	oveq12d 7378	. . 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 7395	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) ∈ V)
12	3, 8, 9, 10, 11	ovmpod 7512	. 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
13	1, 12	eqtrid 2784	1 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ⟨cop 4587  (class class class)co 7360   ∈ cmpo 7362  Catccat 17591   1^st_F c1stf 18096   curry_F ccurf 18137  Δ_funccdiag 18139

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 5242  ax-nul 5252  ax-pr 5378

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 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-diag 18143

This theorem is referenced by:  diagcl  18168  diag11  18170  diag12  18171  diag2  18172  diagpropd  49573