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Theorem curf2val 17350
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 (𝜑𝐶 ∈ Cat)
curf2.d (𝜑𝐷 ∈ Cat)
curf2.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curf2.b 𝐵 = (Base‘𝐷)
curf2.h 𝐻 = (Hom ‘𝐶)
curf2.i 𝐼 = (Id‘𝐷)
curf2.x (𝜑𝑋𝐴)
curf2.y (𝜑𝑌𝐴)
curf2.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
curf2.l 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
curf2.z (𝜑𝑍𝐵)
Assertion
Ref Expression
curf2val (𝜑 → (𝐿𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))

Proof of Theorem curf2val
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 curf2.g . . 3 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 curf2.a . . 3 𝐴 = (Base‘𝐶)
3 curf2.c . . 3 (𝜑𝐶 ∈ Cat)
4 curf2.d . . 3 (𝜑𝐷 ∈ Cat)
5 curf2.f . . 3 (𝜑𝐹 ∈ ((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 (𝜑𝑋𝐴)
10 curf2.y . . 3 (𝜑𝑌𝐴)
11 curf2.k . . 3 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
12 curf2.l . . 3 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12curf2 17349 . 2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 simpr 477 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1514opeq2d 4680 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
1614opeq2d 4680 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
1715, 16oveq12d 6992 . . 3 ((𝜑𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩))
18 eqidd 2772 . . 3 ((𝜑𝑧 = 𝑍) → 𝐾 = 𝐾)
1914fveq2d 6500 . . 3 ((𝜑𝑧 = 𝑍) → (𝐼𝑧) = (𝐼𝑍))
2017, 18, 19oveq123d 6995 . 2 ((𝜑𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
21 curf2.z . 2 (𝜑𝑍𝐵)
22 ovexd 7008 . 2 (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)) ∈ V)
2313, 20, 21, 22fvmptd 6599 1 (𝜑 → (𝐿𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3408  cop 4441  cfv 6185  (class class class)co 6974  2nd c2nd 7498  Basecbs 16337  Hom chom 16430  Catccat 16805  Idccid 16806   Func cfunc 16994   ×c cxpc 17288   curryF ccurf 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-curf 17334
This theorem is referenced by:  curf2cl  17351  curfcl  17352  uncfcurf  17359  yon2  17386
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