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Theorem curf12 18190
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 18188 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6898 . . . . 5 𝐵 ∈ V
1312mptex 7219 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 8062 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op2ndd 7982 . . 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 480 . . 3 ((𝜑𝑦 = 𝑌) → 𝑍𝐵)
20 ovex 7437 . . . . 5 (𝑦𝐽𝑧) ∈ V
2120mptex 7219 . . . 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 480 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25 simprl 768 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑦 = 𝑌)
26 simprr 770 . . . . . 6 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝑧 = 𝑍)
2725, 26oveq12d 7422 . . . . 5 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
2824, 27eleqtrrd 2830 . . . 4 ((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29 ovexd 7439 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30 simplrl 774 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
3130opeq2d 4875 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32 simplrr 775 . . . . . . 7 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
3332opeq2d 4875 . . . . . 6 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
3431, 33oveq12d 7422 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩))
35 eqidd 2727 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1𝑋) = ( 1𝑋))
36 simpr 484 . . . . 5 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
3734, 35, 36oveq123d 7425 . . . 4 (((𝜑 ∧ (𝑦 = 𝑌𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))
3828, 29, 37fvmptdv2 7009 . . 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 7560 . 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 395   = wceq 1533  wcel 2098  Vcvv 3468  cop 4629  cmpt 5224  cfv 6536  (class class class)co 7404  cmpo 7406  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  Hom chom 17215  Catccat 17615  Idccid 17616   Func cfunc 17811   ×c cxpc 18130   curryF ccurf 18173
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-curf 18177
This theorem is referenced by:  curf1cl  18191  curf2cl  18194  uncfcurf  18202  diag12  18207  yon12  18228
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