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Theorem curf2val 18183
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 18182 . 2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
14 simpr 486 . . . . 5 ((𝜑𝑧 = 𝑍) → 𝑧 = 𝑍)
1514opeq2d 4881 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
1614opeq2d 4881 . . . 4 ((𝜑𝑧 = 𝑍) → ⟨𝑌, 𝑧⟩ = ⟨𝑌, 𝑍⟩)
1715, 16oveq12d 7427 . . 3 ((𝜑𝑧 = 𝑍) → (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩) = (⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩))
18 eqidd 2734 . . 3 ((𝜑𝑧 = 𝑍) → 𝐾 = 𝐾)
1914fveq2d 6896 . . 3 ((𝜑𝑧 = 𝑍) → (𝐼𝑧) = (𝐼𝑍))
2017, 18, 19oveq123d 7430 . 2 ((𝜑𝑧 = 𝑍) → (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))
21 curf2.z . 2 (𝜑𝑍𝐵)
22 ovexd 7444 . 2 (𝜑 → (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)) ∈ V)
2313, 20, 21, 22fvmptd 7006 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 4635  cfv 6544  (class class class)co 7409  2nd c2nd 7974  Basecbs 17144  Hom chom 17208  Catccat 17608  Idccid 17609   Func cfunc 17804   ×c cxpc 18120   curryF ccurf 18163
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-curf 18167
This theorem is referenced by:  curf2cl  18184  curfcl  18185  uncfcurf  18192  yon2  18219
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