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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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