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Theorem curf2 17770
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 2739 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
9 eqid 2739 . . . . 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 17764 . . . 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 6753 . . . . . 6 𝐴 ∈ V
1413mptex 7061 . . . . 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 7872 . . . . 5 (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) ∈ V
1614, 15op2ndd 7794 . . . 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 484 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐴)
21 ovex 7268 . . . . . 6 (𝑥𝐻𝑦) ∈ V
2221mptex 7061 . . . . 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 484 . . . . . 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 7253 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2925, 28eleqtrrd 2843 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
307fvexi 6753 . . . . . . 7 𝐵 ∈ V
3130mptex 7061 . . . . . 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 4807 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
35 simplrr 778 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
3635opeq1d 4807 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
3734, 36oveq12d 7253 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩) = (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩))
38 simpr 488 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑔 = 𝐾)
39 eqidd 2740 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼𝑧) = (𝐼𝑧))
4037, 38, 39oveq123d 7256 . . . . . 6 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
4140mpteq2dv 5168 . . . . 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 6858 . . . 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 7388 . . 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 2791 1 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2112  Vcvv 3423  cop 4564  cmpt 5152  cfv 6401  (class class class)co 7235  cmpo 7237  1st c1st 7781  2nd c2nd 7782  Basecbs 16793  Hom chom 16846  Catccat 17200  Idccid 17201   Func cfunc 17393   ×c cxpc 17708   curryF ccurf 17751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5196  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-id 5472  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-f1 6406  df-fo 6407  df-f1o 6408  df-fv 6409  df-ov 7238  df-oprab 7239  df-mpo 7240  df-1st 7783  df-2nd 7784  df-curf 17755
This theorem is referenced by:  curf2val  17771  curf2cl  17772  curfcl  17773  diag2  17786  curf2ndf  17788
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