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Theorem curf2ndf 18204
Description: As shown in diagval 18197, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is 𝑥𝐶 ↦ (𝑦𝐷𝑦), which is a constant functor of the identity functor at 𝐷. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
curf2ndf.q 𝑄 = (𝐷 FuncCat 𝐷)
curf2ndf.c (𝜑𝐶 ∈ Cat)
curf2ndf.d (𝜑𝐷 ∈ Cat)
Assertion
Ref Expression
curf2ndf (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))

Proof of Theorem curf2ndf
Dummy variables 𝑢 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7414 . . . . . . . . . . 11 (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦) = ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑥, 𝑦⟩)
2 eqid 2730 . . . . . . . . . . . . 13 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
3 eqid 2730 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2730 . . . . . . . . . . . . . 14 (Base‘𝐷) = (Base‘𝐷)
52, 3, 4xpcbas 18134 . . . . . . . . . . . . 13 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×c 𝐷))
6 eqid 2730 . . . . . . . . . . . . 13 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
7 curf2ndf.c . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Cat)
87ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
9 curf2ndf.d . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ Cat)
109ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
11 eqid 2730 . . . . . . . . . . . . 13 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
12 opelxpi 5712 . . . . . . . . . . . . . 14 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
1312adantll 710 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
142, 5, 6, 8, 10, 11, 132ndf1 18151 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
15 vex 3476 . . . . . . . . . . . . 13 𝑥 ∈ V
16 vex 3476 . . . . . . . . . . . . 13 𝑦 ∈ V
1715, 16op2nd 7986 . . . . . . . . . . . 12 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
1814, 17eqtrdi 2786 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑥, 𝑦⟩) = 𝑦)
191, 18eqtrid 2782 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦) = 𝑦)
2019mpteq2dva 5247 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ 𝑦))
21 mptresid 6049 . . . . . . . . 9 ( I ↾ (Base‘𝐷)) = (𝑦 ∈ (Base‘𝐷) ↦ 𝑦)
2220, 21eqtr4di 2788 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = ( I ↾ (Base‘𝐷)))
23 df-ov 7414 . . . . . . . . . . . . . . 15 (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓) = ((⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩)
248ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
2510ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
2613ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
27 simp-4r 780 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
28 simplr 765 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
2927, 28opelxpd 5714 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨𝑥, 𝑧⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
302, 5, 6, 24, 25, 11, 26, 292ndf2 18152 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩) = (2nd ↾ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩)))
3130fveq1d 6892 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩) = ((2nd ↾ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩))‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩))
3223, 31eqtrid 2782 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓) = ((2nd ↾ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩))‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩))
33 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (Hom ‘𝐶) = (Hom ‘𝐶)
34 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (Id‘𝐶) = (Id‘𝐶)
357adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
36 simpr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
373, 33, 34, 35, 36catidcl 17630 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
3837ad5ant12 752 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
39 simpr 483 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧))
4038, 39opelxpd 5714 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨((Id‘𝐶)‘𝑥), 𝑓⟩ ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧)))
41 eqid 2730 . . . . . . . . . . . . . . . . . 18 (Hom ‘𝐷) = (Hom ‘𝐷)
42 simpllr 772 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
432, 3, 4, 33, 41, 27, 42, 27, 28, 6xpchom2 18142 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩) = ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧)))
4440, 43eleqtrrd 2834 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ⟨((Id‘𝐶)‘𝑥), 𝑓⟩ ∈ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩))
4544fvresd 6910 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩))‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩) = (2nd ‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩))
46 fvex 6903 . . . . . . . . . . . . . . . 16 ((Id‘𝐶)‘𝑥) ∈ V
47 vex 3476 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
4846, 47op2nd 7986 . . . . . . . . . . . . . . 15 (2nd ‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩) = 𝑓
4945, 48eqtrdi 2786 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾ (⟨𝑥, 𝑦⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑥, 𝑧⟩))‘⟨((Id‘𝐶)‘𝑥), 𝑓⟩) = 𝑓)
5032, 49eqtrd 2770 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓) = 𝑓)
5150mpteq2dva 5247 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓)) = (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓))
52 mptresid 6049 . . . . . . . . . . . 12 ( I ↾ (𝑦(Hom ‘𝐷)𝑧)) = (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓)
5351, 52eqtr4di 2788 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))
54533impa 1108 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))
5554mpoeq3dva 7488 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧))))
56 fveq2 6890 . . . . . . . . . . . 12 (𝑢 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨𝑦, 𝑧⟩))
57 df-ov 7414 . . . . . . . . . . . 12 (𝑦(Hom ‘𝐷)𝑧) = ((Hom ‘𝐷)‘⟨𝑦, 𝑧⟩)
5856, 57eqtr4di 2788 . . . . . . . . . . 11 (𝑢 = ⟨𝑦, 𝑧⟩ → ((Hom ‘𝐷)‘𝑢) = (𝑦(Hom ‘𝐷)𝑧))
5958reseq2d 5980 . . . . . . . . . 10 (𝑢 = ⟨𝑦, 𝑧⟩ → ( I ↾ ((Hom ‘𝐷)‘𝑢)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))
6059mpompt 7524 . . . . . . . . 9 (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))
6155, 60eqtr4di 2788 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢))))
6222, 61opeq12d 4880 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓)))⟩ = ⟨( I ↾ (Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))⟩)
63 eqid 2730 . . . . . . . 8 (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))
649adantr 479 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
652, 7, 9, 112ndfcl 18154 . . . . . . . . 9 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
6665adantr 479 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
67 eqid 2730 . . . . . . . 8 ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥)
6863, 3, 35, 64, 66, 4, 36, 67, 41, 34curf1 18182 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥) = ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑥, 𝑧⟩)𝑓)))⟩)
69 eqid 2730 . . . . . . . 8 (idfunc𝐷) = (idfunc𝐷)
7069, 4, 64, 41idfuval 17830 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (idfunc𝐷) = ⟨( I ↾ (Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))⟩)
7162, 68, 703eqtr4d 2780 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥) = (idfunc𝐷))
72 eqid 2730 . . . . . . 7 (𝑄Δfunc𝐶) = (𝑄Δfunc𝐶)
73 curf2ndf.q . . . . . . . . 9 𝑄 = (𝐷 FuncCat 𝐷)
7473, 9, 9fuccat 17927 . . . . . . . 8 (𝜑𝑄 ∈ Cat)
7574adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑄 ∈ Cat)
7673fucbas 17916 . . . . . . 7 (𝐷 Func 𝐷) = (Base‘𝑄)
7769idfucl 17835 . . . . . . . . 9 (𝐷 ∈ Cat → (idfunc𝐷) ∈ (𝐷 Func 𝐷))
789, 77syl 17 . . . . . . . 8 (𝜑 → (idfunc𝐷) ∈ (𝐷 Func 𝐷))
7978adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (idfunc𝐷) ∈ (𝐷 Func 𝐷))
80 eqid 2730 . . . . . . 7 ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))
8172, 75, 35, 76, 79, 80, 3, 36diag11 18200 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑥) = (idfunc𝐷))
8271, 81eqtr4d 2773 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥) = ((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑥))
8382mpteq2dva 5247 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑥)))
84 relfunc 17816 . . . . . . 7 Rel (𝐶 Func 𝑄)
8563, 73, 7, 9, 65curfcl 18189 . . . . . . 7 (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) ∈ (𝐶 Func 𝑄))
86 1st2ndbr 8030 . . . . . . 7 ((Rel (𝐶 Func 𝑄) ∧ (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))))
8784, 85, 86sylancr 585 . . . . . 6 (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))))
883, 76, 87funcf1 17820 . . . . 5 (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷))
8988feqmptd 6959 . . . 4 (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥)))
9072, 74, 7, 76, 78, 80diag1cl 18199 . . . . . . 7 (𝜑 → ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) ∈ (𝐶 Func 𝑄))
91 1st2ndbr 8030 . . . . . . 7 ((Rel (𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) ∈ (𝐶 Func 𝑄)) → (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))))
9284, 90, 91sylancr 585 . . . . . 6 (𝜑 → (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))))
933, 76, 92funcf1 17820 . . . . 5 (𝜑 → (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷))
9493feqmptd 6959 . . . 4 (𝜑 → (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑥)))
9583, 89, 943eqtr4d 2780 . . 3 (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) = (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))))
969ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
9769, 4, 96idfu1st 17833 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘(idfunc𝐷)) = ( I ↾ (Base‘𝐷)))
9897coeq2d 5861 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ (1st ‘(idfunc𝐷))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))))
99 eqid 2730 . . . . . . . . . . 11 (Id‘𝑄) = (Id‘𝑄)
100 eqid 2730 . . . . . . . . . . 11 (Id‘𝐷) = (Id‘𝐷)
10178ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (idfunc𝐷) ∈ (𝐷 Func 𝐷))
10273, 99, 100, 101fucid 17928 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝑄)‘(idfunc𝐷)) = ((Id‘𝐷) ∘ (1st ‘(idfunc𝐷))))
1034, 100cidfn 17627 . . . . . . . . . . . . . 14 (𝐷 ∈ Cat → (Id‘𝐷) Fn (Base‘𝐷))
10496, 103syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) Fn (Base‘𝐷))
105 dffn2 6718 . . . . . . . . . . . . 13 ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
106104, 105sylib 217 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷):(Base‘𝐷)⟶V)
107106feqmptd 6959 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧)))
108 fcoi1 6764 . . . . . . . . . . . 12 ((Id‘𝐷):(Base‘𝐷)⟶V → ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))) = (Id‘𝐷))
109106, 108syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))) = (Id‘𝐷))
1107ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
111110adantr 479 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
11296adantr 479 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
113 simplrl 773 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
114 opelxpi 5712 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑧⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
115113, 114sylan 578 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑥, 𝑧⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
116 simplrr 774 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
117 opelxpi 5712 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑧⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
118116, 117sylan 578 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑦, 𝑧⟩ ∈ ((Base‘𝐶) × (Base‘𝐷)))
1192, 5, 6, 111, 112, 11, 115, 1182ndf2 18152 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩) = (2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩)))
120119oveqd 7428 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = (𝑓(2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))((Id‘𝐷)‘𝑧)))
121 df-ov 7414 . . . . . . . . . . . . . . 15 (𝑓(2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))((Id‘𝐷)‘𝑧)) = ((2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))‘⟨𝑓, ((Id‘𝐷)‘𝑧)⟩)
122 simplr 765 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
123 simpr 483 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷))
1244, 41, 100, 112, 123catidcl 17630 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
125122, 124opelxpd 5714 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑧)⟩ ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧)))
126113adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
127116adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
1282, 3, 4, 33, 41, 126, 123, 127, 123, 6xpchom2 18142 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩) = ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧)))
129125, 128eleqtrrd 2834 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ⟨𝑓, ((Id‘𝐷)‘𝑧)⟩ ∈ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))
130129fvresd 6910 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))‘⟨𝑓, ((Id‘𝐷)‘𝑧)⟩) = (2nd ‘⟨𝑓, ((Id‘𝐷)‘𝑧)⟩))
131121, 130eqtrid 2782 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))((Id‘𝐷)‘𝑧)) = (2nd ‘⟨𝑓, ((Id‘𝐷)‘𝑧)⟩))
132 fvex 6903 . . . . . . . . . . . . . . 15 ((Id‘𝐷)‘𝑧) ∈ V
13347, 132op2nd 7986 . . . . . . . . . . . . . 14 (2nd ‘⟨𝑓, ((Id‘𝐷)‘𝑧)⟩) = ((Id‘𝐷)‘𝑧)
134131, 133eqtrdi 2786 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (⟨𝑥, 𝑧⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑦, 𝑧⟩))((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧))
135120, 134eqtrd 2770 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧))
136135mpteq2dva 5247 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧)))
137107, 109, 1363eqtr4rd 2781 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))))
13898, 102, 1373eqtr4rd 2781 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = ((Id‘𝑄)‘(idfunc𝐷)))
13965ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
140 simpr 483 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
141 eqid 2730 . . . . . . . . . 10 ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓)
14263, 3, 110, 96, 139, 4, 33, 100, 113, 116, 140, 141curf2 18186 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓) = (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(⟨𝑥, 𝑧⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))
14374ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑄 ∈ Cat)
14472, 143, 110, 76, 101, 80, 3, 113, 33, 99, 116, 140diag12 18201 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)‘𝑓) = ((Id‘𝑄)‘(idfunc𝐷)))
145138, 142, 1443eqtr4d 2780 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)‘𝑓))
146145mpteq2dva 5247 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)‘𝑓)))
147 eqid 2730 . . . . . . . . . 10 (𝐷 Nat 𝐷) = (𝐷 Nat 𝐷)
14873, 147fuchom 17917 . . . . . . . . 9 (𝐷 Nat 𝐷) = (Hom ‘𝑄)
14987adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))))
150 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
151 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
1523, 33, 148, 149, 150, 151funcf2 17822 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))‘𝑦)))
153152feqmptd 6959 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)‘𝑓)))
15492adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))))
1553, 33, 148, 154, 150, 151funcf2 17822 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))‘𝑦)))
156155feqmptd 6959 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)‘𝑓)))
157146, 153, 1563eqtr4d 2780 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦))
1581573impb 1113 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦))
159158mpoeq3dva 7488 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)))
1603, 87funcfn2 17823 . . . . 5 (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)))
161 fnov 7542 . . . . 5 ((2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)))
162160, 161sylib 217 . . . 4 (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))𝑦)))
1633, 92funcfn2 17823 . . . . 5 (𝜑 → (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)))
164 fnov 7542 . . . . 5 ((2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)))
165163, 164sylib 217 . . . 4 (𝜑 → (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))𝑦)))
166159, 162, 1653eqtr4d 2780 . . 3 (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))) = (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))))
16795, 166opeq12d 4880 . 2 (𝜑 → ⟨(1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))), (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))⟩ = ⟨(1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))), (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))⟩)
168 1st2nd 8027 . . 3 ((Rel (𝐶 Func 𝑄) ∧ (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))), (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))⟩)
16984, 85, 168sylancr 585 . 2 (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷))), (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)))⟩)
170 1st2nd 8027 . . 3 ((Rel (𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) ∈ (𝐶 Func 𝑄)) → ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) = ⟨(1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))), (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))⟩)
17184, 90, 170sylancr 585 . 2 (𝜑 → ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)) = ⟨(1st ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷))), (2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))⟩)
172167, 169, 1713eqtr4d 2780 1 (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  cop 4633   class class class wbr 5147  cmpt 5230   I cid 5572   × cxp 5673  cres 5677  ccom 5679  Rel wrel 5680   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976  Basecbs 17148  Hom chom 17212  Catccat 17612  Idccid 17613   Func cfunc 17808  idfunccidfu 17809   Nat cnat 17896   FuncCat cfuc 17897   ×c cxpc 18124   2ndF c2ndf 18126   curryF ccurf 18167  Δfunccdiag 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-hom 17225  df-cco 17226  df-cat 17616  df-cid 17617  df-func 17812  df-idfu 17813  df-nat 17898  df-fuc 17899  df-xpc 18128  df-1stf 18129  df-2ndf 18130  df-curf 18171  df-diag 18173
This theorem is referenced by: (None)
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