Theorem diagval 17490

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 17466	. . . 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 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	opeq12d 4811	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
7	4, 5	oveq12d 7174	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷))
8	6, 7	oveq12d 7174	. . 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 7191	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) ∈ V)
12	3, 8, 9, 10, 11	ovmpod 7302	. 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
13	1, 12	syl5eq 2868	1 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573  (class class class)co 7156   ∈ cmpo 7158  Catccat 16935   1^st_F c1stf 17419   curry_F ccurf 17460  Δ_funccdiag 17462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-diag 17466

This theorem is referenced by:  diagcl  17491  diag11  17493  diag12  17494  diag2  17495