Theorem curf2val 18286

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 18285	. 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 4884	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
16	14	opeq2d 4884	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
17	15, 16	oveq12d 7448	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩))
18		eqidd 2735	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 = 𝐾)
19	14	fveq2d 6910	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝐼‘𝑧) = (𝐼‘𝑍))
20	17, 18, 19	oveq123d 7451	. 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 7465	. 2 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)) ∈ V)
23	13, 20, 21, 22	fvmptd 7022	1 ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  Vcvv 3477  ⟨cop 4636  ‘cfv 6562  (class class class)co 7430  2^nd c2nd 8011  Basecbs 17244  Hom chom 17308  Catccat 17708  Idccid 17709   Func cfunc 17904   ×_c cxpc 18223   curry_F ccurf 18266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-curf 18270

This theorem is referenced by:  curf2cl  18287  curfcl  18288  uncfcurf  18295  yon2  18322