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Theorem curf12 18224
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 18222 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6914 . . . . 5 𝐵 ∈ V
1312mptex 7239 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 8088 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op2ndd 8008 . . 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 479 . . 3 ((𝜑𝑦 = 𝑌) → 𝑍𝐵)
20 ovex 7457 . . . . 5 (𝑦𝐽𝑧) ∈ V
2120mptex 7239 . . . 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 479 . . . . 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 7442 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
2824, 27eleqtrrd 2831 . . . 4 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29 ovexd 7459 . . . 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 4883 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
3332opeq2d 4883 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
3431, 33oveq12d 7442 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩))
35 eqidd 2728 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1𝑋) = ( 1𝑋))
36 simpr 483 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
3734, 35, 36oveq123d 7445 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))
3828, 29, 37fvmptdv2 7026 . . 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 7582 . 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 394   = wceq 1533  wcel 2098  Vcvv 3471  cop 4636  cmpt 5233  cfv 6551  (class class class)co 7424  cmpo 7426  1st c1st 7995  2nd c2nd 7996  Basecbs 17185  Hom chom 17249  Catccat 17649  Idccid 17650   Func cfunc 17845   ×c cxpc 18164   curryF ccurf 18207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-curf 18211
This theorem is referenced by:  curf1cl  18225  curf2cl  18228  uncfcurf  18236  diag12  18241  yon12  18262
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