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Theorem curf2val 18045
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 18044 . 2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 simpr 485 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1514opeq2d 4824 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
1614opeq2d 4824 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
1715, 16oveq12d 7355 . . 3 ((𝜑𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩))
18 eqidd 2737 . . 3 ((𝜑𝑧 = 𝑍) → 𝐾 = 𝐾)
1914fveq2d 6829 . . 3 ((𝜑𝑧 = 𝑍) → (𝐼𝑧) = (𝐼𝑍))
2017, 18, 19oveq123d 7358 . 2 ((𝜑𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
21 curf2.z . 2 (𝜑𝑍𝐵)
22 ovexd 7372 . 2 (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)) ∈ V)
2313, 20, 21, 22fvmptd 6938 1 (𝜑 → (𝐿𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  cop 4579  cfv 6479  (class class class)co 7337  2nd c2nd 7898  Basecbs 17009  Hom chom 17070  Catccat 17470  Idccid 17471   Func cfunc 17666   ×c cxpc 17982   curryF ccurf 18025
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-curf 18029
This theorem is referenced by:  curf2cl  18046  curfcl  18047  uncfcurf  18054  yon2  18081
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