Theorem curf2val 18303

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 18302	. 2 ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
14		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
15	14	opeq2d 4847	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
16	14	opeq2d 4847	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
17	15, 16	oveq12d 7434	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩))
18		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾)
19	14	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍))
20	17, 18, 19	oveq123d 7437	. 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 7451	. 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V)
23	13, 20, 21, 22	fvmptd 7001	1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597  ‘cfv 6540  (class class class)co 7416  2^nd c2nd 7987  Basecbs 17286  Hom chom 17338  Catccat 17737  Idccid 17738   Func cfunc 17928   ×_c cxpc 18241   curry_F ccurf 18283

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-curf 18287

This theorem is used by:  curf2cl  18304  curfcl  18305  uncfcurf  18312  yon2  18339