Theorem diagval 18328

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 18304	. . . 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 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑐 = 𝐶)
5		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → 𝑑 = 𝐷)
6	4, 5	opeq12d 4841	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
7	4, 5	oveq12d 7431	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (𝑐 1stF 𝑑) = (𝐶 1stF 𝐷))
8	6, 7	oveq12d 7431	. . 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 7448	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) ∈ V)
12	3, 8, 9, 10, 11	ovmpod 7565	. 2 ⊢ (𝜑 → (𝐶Δfunc𝐷) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
13	1, 12	eqtrid 2807	1 ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590  (class class class)co 7413   ∈ cmpo 7415  Catccat 17752   1^st_F c1stf 18257   curry_F ccurf 18298  Δ_funccdiag 18300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-diag 18304

This theorem is used by:  diagcl  18329  diag11  18331  diag12  18332  diag2  18333  diagpropd  50218