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Theorem curf12 18043
Description: The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
curfval.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a 𝐴 = (Base‘𝐶)
curfval.c (𝜑𝐶 ∈ Cat)
curfval.d (𝜑𝐷 ∈ Cat)
curfval.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b 𝐵 = (Base‘𝐷)
curf1.x (𝜑𝑋𝐴)
curf1.k 𝐾 = ((1st𝐺)‘𝑋)
curf11.y (𝜑𝑌𝐵)
curf12.j 𝐽 = (Hom ‘𝐷)
curf12.1 1 = (Id‘𝐶)
curf12.y (𝜑𝑍𝐵)
curf12.g (𝜑𝐻 ∈ (𝑌𝐽𝑍))
Assertion
Ref Expression
curf12 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐻) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))

Proof of Theorem curf12
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 curfval.a . . . 4 𝐴 = (Base‘𝐶)
3 curfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 curfval.d . . . 4 (𝜑𝐷 ∈ Cat)
5 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6 curfval.b . . . 4 𝐵 = (Base‘𝐷)
7 curf1.x . . . 4 (𝜑𝑋𝐴)
8 curf1.k . . . 4 𝐾 = ((1st𝐺)‘𝑋)
9 curf12.j . . . 4 𝐽 = (Hom ‘𝐷)
10 curf12.1 . . . 4 1 = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 18041 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6844 . . . . 5 𝐵 ∈ V
1312mptex 7160 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 7993 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op2ndd 7915 . . 3 (𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
1611, 15syl 17 . 2 (𝜑 → (2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))))
17 curf11.y . . 3 (𝜑𝑌𝐵)
18 curf12.y . . . 4 (𝜑𝑍𝐵)
1918adantr 482 . . 3 ((𝜑𝑦 = 𝑌) → 𝑍𝐵)
20 ovex 7375 . . . . 5 (𝑦𝐽𝑧) ∈ V
2120mptex 7160 . . . 4 (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
2221a1i 11 . . 3 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
23 curf12.g . . . . . 6 (𝜑𝐻 ∈ (𝑌𝐽𝑍))
2423adantr 482 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑦 = 𝑌)
26 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑧 = 𝑍)
2725, 26oveq12d 7360 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
2824, 27eleqtrrd 2841 . . . 4 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29 ovexd 7377 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30 simplrl 775 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
3130opeq2d 4829 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
3332opeq2d 4829 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
3431, 33oveq12d 7360 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩))
35 eqidd 2738 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1𝑋) = ( 1𝑋))
36 simpr 486 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
3734, 35, 36oveq123d 7363 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))
3828, 29, 37fvmptdv2 6954 . . 3 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → ((𝑌(2nd𝐾)𝑍) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)) → ((𝑌(2nd𝐾)𝑍)‘𝐻) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻)))
3917, 19, 22, 38ovmpodv 7497 . 2 (𝜑 → ((2nd𝐾) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) → ((𝑌(2nd𝐾)𝑍)‘𝐻) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻)))
4016, 39mpd 15 1 (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐻) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  Vcvv 3442  cop 4584  cmpt 5180  cfv 6484  (class class class)co 7342  cmpo 7344  1st c1st 7902  2nd c2nd 7903  Basecbs 17010  Hom chom 17071  Catccat 17471  Idccid 17472   Func cfunc 17667   ×c cxpc 17983   curryF ccurf 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-1st 7904  df-2nd 7905  df-curf 18030
This theorem is referenced by:  curf1cl  18044  curf2cl  18047  uncfcurf  18055  diag12  18060  yon12  18081
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