Theorem curf2val 18146

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 18145	. 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 4833	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
16	14	opeq2d 4833	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
17	15, 16	oveq12d 7373	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩))
18		eqidd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾)
19	14	fveq2d 6835	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍))
20	17, 18, 19	oveq123d 7376	. 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 7390	. 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V)
23	13, 20, 21, 22	fvmptd 6945	1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355  2^nd c2nd 7929  Basecbs 17130  Hom chom 17182  Catccat 17580  Idccid 17581   Func cfunc 17771   ×_c cxpc 18084   curry_F ccurf 18126

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-curf 18130

This theorem is referenced by:  curf2cl  18147  curfcl  18148  uncfcurf  18155  yon2  18182