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Theorem curf2val 18180
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 18179 . 2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 simpr 486 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1514opeq2d 4880 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
1614opeq2d 4880 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
1715, 16oveq12d 7424 . . 3 ((𝜑𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩))
18 eqidd 2734 . . 3 ((𝜑𝑧 = 𝑍) → 𝐾 = 𝐾)
1914fveq2d 6893 . . 3 ((𝜑𝑧 = 𝑍) → (𝐼𝑧) = (𝐼𝑍))
2017, 18, 19oveq123d 7427 . 2 ((𝜑𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
21 curf2.z . 2 (𝜑𝑍𝐵)
22 ovexd 7441 . 2 (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)) ∈ V)
2313, 20, 21, 22fvmptd 7003 1 (𝜑 → (𝐿𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cop 4634  cfv 6541  (class class class)co 7406  2nd c2nd 7971  Basecbs 17141  Hom chom 17205  Catccat 17605  Idccid 17606   Func cfunc 17801   ×c cxpc 18117   curryF ccurf 18160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-curf 18164
This theorem is referenced by:  curf2cl  18181  curfcl  18182  uncfcurf  18189  yon2  18216
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