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Theorem curf2 17266
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 2778 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
9 eqid 2778 . . . . 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 17260 . . . 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 6462 . . . . . 6 𝐴 ∈ V
1413mptex 6760 . . . . 5 (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
1513, 13mpt2ex 7529 . . . . 5 (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) ∈ V
1614, 15op2ndd 7458 . . . 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 474 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐴)
21 ovex 6956 . . . . . 6 (𝑥𝐻𝑦) ∈ V
2221mptex 6760 . . . . 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 474 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑋𝐻𝑌))
26 simprl 761 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
27 simprr 763 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
2826, 27oveq12d 6942 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2925, 28eleqtrrd 2862 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
307fvexi 6462 . . . . . . 7 𝐵 ∈ V
3130mptex 6760 . . . . . 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 767 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑥 = 𝑋)
3433opeq1d 4644 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
35 simplrr 768 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
3635opeq1d 4644 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
3734, 36oveq12d 6942 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩) = (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩))
38 simpr 479 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑔 = 𝐾)
39 eqidd 2779 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼𝑧) = (𝐼𝑧))
4037, 38, 39oveq123d 6945 . . . . . 6 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
4140mpteq2dv 4982 . . . . 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 6561 . . . 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, 42ovmpt2dv 7072 . . 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, 44syl5eq 2826 1 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cop 4404  cmpt 4967  cfv 6137  (class class class)co 6924  cmpt2 6926  1st c1st 7445  2nd c2nd 7446  Basecbs 16266  Hom chom 16360  Catccat 16721  Idccid 16722   Func cfunc 16910   ×c cxpc 17205   curryF ccurf 17247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-curf 17251
This theorem is referenced by:  curf2val  17267  curf2cl  17268  curfcl  17269  diag2  17282  curf2ndf  17284
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