Theorem curf2val 18275

Description: Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curf2.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curf2.a	⊢ 𝐴 = (Base‘𝐶)
curf2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curf2.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curf2.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curf2.b	⊢ 𝐵 = (Base‘𝐷)
curf2.h	⊢ 𝐻 = (Hom ‘𝐶)
curf2.i	⊢ 𝐼 = (Id‘𝐷)
curf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
curf2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
curf2.l	⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)
curf2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
curf2val	⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))

Proof of Theorem curf2val

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf2.g	. . 3 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		curf2.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
3		curf2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curf2.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curf2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		curf2.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
7		curf2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
8		curf2.i	. . 3 ⊢ 𝐼 = (Id‘𝐷)
9		curf2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
10		curf2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
11		curf2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
12		curf2.l	. . 3 ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	curf2 18274	. 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
14		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
15	14	opeq2d 4880	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
16	14	opeq2d 4880	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
17	15, 16	oveq12d 7449	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩))
18		eqidd 2738	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾)
19	14	fveq2d 6910	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍))
20	17, 18, 19	oveq123d 7452	. 2 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))
21		curf2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22		ovexd 7466	. 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V)
23	13, 20, 21, 22	fvmptd 7023	1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632  ‘cfv 6561  (class class class)co 7431  2^nd c2nd 8013  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708   Func cfunc 17899   ×_c cxpc 18213   curry_F ccurf 18255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-curf 18259

This theorem is referenced by:  curf2cl  18276  curfcl  18277  uncfcurf  18284  yon2  18311