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Theorem curf12 17469
 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 17467 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6659 . . . . 5 𝐵 ∈ V
1312mptex 6963 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 7760 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op2ndd 7682 . . 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 484 . . 3 ((𝜑𝑦 = 𝑌) → 𝑍𝐵)
20 ovex 7168 . . . . 5 (𝑦𝐽𝑧) ∈ V
2120mptex 6963 . . . 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 484 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑦 = 𝑌)
26 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑧 = 𝑍)
2725, 26oveq12d 7153 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
2824, 27eleqtrrd 2893 . . . 4 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29 ovexd 7170 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30 simplrl 776 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
3130opeq2d 4772 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32 simplrr 777 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
3332opeq2d 4772 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
3431, 33oveq12d 7153 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩))
35 eqidd 2799 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1𝑋) = ( 1𝑋))
36 simpr 488 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
3734, 35, 36oveq123d 7156 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))
3828, 29, 37fvmptdv2 6763 . . 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 7286 . 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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1st c1st 7669  2nd c2nd 7670  Basecbs 16475  Hom chom 16568  Catccat 16927  Idccid 16928   Func cfunc 17116   ×c cxpc 17410   curryF ccurf 17452 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-curf 17456 This theorem is referenced by:  curf1cl  17470  curf2cl  17473  uncfcurf  17481  diag12  17486  yon12  17507
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