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Theorem curf2 16641
Description: Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curf2.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curf2.a 𝐴 = (Base‘𝐶)
curf2.c (𝜑𝐶 ∈ Cat)
curf2.d (𝜑𝐷 ∈ Cat)
curf2.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curf2.b 𝐵 = (Base‘𝐷)
curf2.h 𝐻 = (Hom ‘𝐶)
curf2.i 𝐼 = (Id‘𝐷)
curf2.x (𝜑𝑋𝐴)
curf2.y (𝜑𝑌𝐴)
curf2.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
curf2.l 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
Assertion
Ref Expression
curf2 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Distinct variable groups:   𝑧,𝐶   𝑧,𝐹   𝑧,𝐻   𝑧,𝐿   𝑧,𝐸   𝑧,𝐺   𝑧,𝐼   𝜑,𝑧   𝑧,𝐵   𝑧,𝐷   𝑧,𝑋   𝑧,𝐾   𝑧,𝑌
Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem curf2
Dummy variables 𝑥 𝑦 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.l . 2 𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)
2 curf2.g . . . . 5 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3 curf2.a . . . . 5 𝐴 = (Base‘𝐶)
4 curf2.c . . . . 5 (𝜑𝐶 ∈ Cat)
5 curf2.d . . . . 5 (𝜑𝐷 ∈ Cat)
6 curf2.f . . . . 5 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7 curf2.b . . . . 5 𝐵 = (Base‘𝐷)
8 eqid 2609 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
9 eqid 2609 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
10 curf2.h . . . . 5 𝐻 = (Hom ‘𝐶)
11 curf2.i . . . . 5 𝐼 = (Id‘𝐷)
122, 3, 4, 5, 6, 7, 8, 9, 10, 11curfval 16635 . . . 4 (𝜑𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
13 fvex 6098 . . . . . . 7 (Base‘𝐶) ∈ V
143, 13eqeltri 2683 . . . . . 6 𝐴 ∈ V
1514mptex 6368 . . . . 5 (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) ∈ V
1614, 14mpt2ex 7114 . . . . 5 (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) ∈ V
1715, 16op2ndd 7048 . . . 4 (𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩ → (2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))))
1812, 17syl 17 . . 3 (𝜑 → (2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))))
19 curf2.x . . . 4 (𝜑𝑋𝐴)
20 curf2.y . . . . 5 (𝜑𝑌𝐴)
2120adantr 479 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐴)
22 ovex 6555 . . . . . 6 (𝑥𝐻𝑦) ∈ V
2322mptex 6368 . . . . 5 (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) ∈ V
2423a1i 11 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) ∈ V)
25 curf2.k . . . . . . 7 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
2625adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑋𝐻𝑌))
27 simprl 789 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
28 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
2927, 28oveq12d 6545 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
3026, 29eleqtrrd 2690 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝐾 ∈ (𝑥𝐻𝑦))
31 fvex 6098 . . . . . . . 8 (Base‘𝐷) ∈ V
327, 31eqeltri 2683 . . . . . . 7 𝐵 ∈ V
3332mptex 6368 . . . . . 6 (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) ∈ V
3433a1i 11 . . . . 5 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) ∈ V)
35 simplrl 795 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑥 = 𝑋)
3635opeq1d 4340 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
37 simplrr 796 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑦 = 𝑌)
3837opeq1d 4340 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → ⟨𝑦, 𝑧⟩ = ⟨𝑌, 𝑧⟩)
3936, 38oveq12d 6545 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩) = (⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩))
40 simpr 475 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → 𝑔 = 𝐾)
41 eqidd 2610 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝐼𝑧) = (𝐼𝑧))
4239, 40, 41oveq123d 6548 . . . . . 6 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)) = (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))
4342mpteq2dv 4667 . . . . 5 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑔 = 𝐾) → (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
4430, 34, 43fvmptdv2 6191 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑋(2nd𝐺)𝑌) = (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))) → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))))
4519, 21, 24, 44ovmpt2dv 6669 . . 3 (𝜑 → ((2nd𝐺) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))) → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧)))))
4618, 45mpd 15 . 2 (𝜑 → ((𝑋(2nd𝐺)𝑌)‘𝐾) = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
471, 46syl5eq 2655 1 (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  cmpt 4637  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7035  2nd c2nd 7036  Basecbs 15644  Hom chom 15728  Catccat 16097  Idccid 16098   Func cfunc 16286   ×c cxpc 16580   curryF ccurf 16622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-curf 16626
This theorem is referenced by:  curf2val  16642  curf2cl  16643  curfcl  16644  diag2  16657  curf2ndf  16659
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