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Theorem curf2 18274
Description: Value of the curry functor at a morphism. (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)
Assertion
Ref Expression
curf2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Distinct variable groups:   𝑧,𝐶   𝑧,𝐹   𝑧,𝐻   𝑧,𝐿   𝑧,𝐸   𝑧,𝐺   𝑧,𝐼   𝜑,𝑧   𝑧,𝐵   𝑧,𝐷   𝑧,𝑋   𝑧,𝐾   𝑧,𝑌
Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem curf2
Dummy variables 𝑥 𝑦 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.l . 2 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
2 curf2.g . . . . 5 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3 curf2.a . . . . 5 𝐴 = (Base‘𝐶)
4 curf2.c . . . . 5 (𝜑𝐶 ∈ Cat)
5 curf2.d . . . . 5 (𝜑𝐷 ∈ Cat)
6 curf2.f . . . . 5 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7 curf2.b . . . . 5 𝐵 = (Base‘𝐷)
8 eqid 2737 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
9 eqid 2737 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
10 curf2.h . . . . 5 𝐻 = (Hom ‘𝐶)
11 curf2.i . . . . 5 𝐼 = (Id‘𝐷)
122, 3, 4, 5, 6, 7, 8, 9, 10, 11curfval 18268 . . . 4 (𝜑𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
133fvexi 6920 . . . . . 6 𝐴 ∈ V
1413mptex 7243 . . . . 5 (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
1513, 13mpoex 8104 . . . . 5 (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) ∈ V
1614, 15op2ndd 8025 . . . 4 (𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩ → (2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))))
1712, 16syl 17 . . 3 (𝜑 → (2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))))
18 curf2.x . . . 4 (𝜑𝑋𝐴)
19 curf2.y . . . . 5 (𝜑𝑌𝐴)
2019adantr 480 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐴)
21 ovex 7464 . . . . . 6 (𝑥𝐻𝑦) ∈ V
2221mptex 7243 . . . . 5 (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) ∈ V
2322a1i 11 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) ∈ V)
24 curf2.k . . . . . . 7 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
2524adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑋𝐻𝑌))
26 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
27 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
2826, 27oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2925, 28eleqtrrd 2844 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
307fvexi 6920 . . . . . . 7 𝐵 ∈ V
3130mptex 7243 . . . . . 6 (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) ∈ V
3231a1i 11 . . . . 5 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) ∈ V)
33 simplrl 777 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑥 = 𝑋)
3433opeq1d 4879 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
35 simplrr 778 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
3635opeq1d 4879 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
3734, 36oveq12d 7449 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩) = (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩))
38 simpr 484 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑔 = 𝐾)
39 eqidd 2738 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼𝑧) = (𝐼𝑧))
4037, 38, 39oveq123d 7452 . . . . . 6 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
4140mpteq2dv 5244 . . . . 5 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
4229, 32, 41fvmptdv2 7034 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑋(2nd𝐺)𝑌) = (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))))
4318, 20, 23, 42ovmpodv 7590 . . 3 (𝜑 → ((2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))))
4417, 43mpd 15 . 2 (𝜑 → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
451, 44eqtrid 2789 1 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708   Func cfunc 17899   ×c cxpc 18213   curryF ccurf 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-curf 18259
This theorem is referenced by:  curf2val  18275  curf2cl  18276  curfcl  18277  diag2  18290  curf2ndf  18292  tposcurf2  49000
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