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Theorem curfval 18104
Description: Value of the curry functor. (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)
curfval.j 𝐽 = (Hom ‘𝐷)
curfval.1 1 = (Id‘𝐶)
curfval.h 𝐻 = (Hom ‘𝐶)
curfval.i 𝐼 = (Id‘𝐷)
Assertion
Ref Expression
curfval (𝜑𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
Distinct variable groups:   𝑥,𝑔,𝑦,𝑧, 1   𝑥,𝐴,𝑦   𝐵,𝑔,𝑥,𝑦,𝑧   𝐶,𝑔,𝑥,𝑦,𝑧   𝐷,𝑔,𝑥,𝑦,𝑧   𝑔,𝐻,𝑦,𝑧   𝜑,𝑔,𝑥,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽,𝑥   𝑔,𝐹,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐸(𝑥)   𝐺(𝑥,𝑦,𝑧,𝑔)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curfval
Dummy variables 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . 2 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 df-curf 18095 . . . 4 curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ (1st𝑒) / 𝑐(2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)
32a1i 11 . . 3 (𝜑 → curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ (1st𝑒) / 𝑐(2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩))
4 fvexd 6854 . . . 4 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) ∈ V)
5 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → 𝑒 = ⟨𝐶, 𝐷⟩)
65fveq2d 6843 . . . . 5 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) = (1st ‘⟨𝐶, 𝐷⟩))
7 curfval.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
8 curfval.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
9 op1stg 7929 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
107, 8, 9syl2anc 584 . . . . . 6 (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
1110adantr 481 . . . . 5 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
126, 11eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) = 𝐶)
13 fvexd 6854 . . . . 5 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) ∈ V)
145adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → 𝑒 = ⟨𝐶, 𝐷⟩)
1514fveq2d 6843 . . . . . 6 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) = (2nd ‘⟨𝐶, 𝐷⟩))
16 op2ndg 7930 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
177, 8, 16syl2anc 584 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
1817ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
1915, 18eqtrd 2776 . . . . 5 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) = 𝐷)
20 simplr 767 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
2120fveq2d 6843 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
22 curfval.a . . . . . . . 8 𝐴 = (Base‘𝐶)
2321, 22eqtr4di 2794 . . . . . . 7 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐴)
24 simpr 485 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
2524fveq2d 6843 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
26 curfval.b . . . . . . . . . 10 𝐵 = (Base‘𝐷)
2725, 26eqtr4di 2794 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑑) = 𝐵)
28 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
2928ad2antrr 724 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑓 = 𝐹)
3029fveq2d 6843 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (1st𝑓) = (1st𝐹))
3130oveqd 7370 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(1st𝑓)𝑦) = (𝑥(1st𝐹)𝑦))
3227, 31mpteq12dv 5194 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)) = (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)))
3324fveq2d 6843 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
34 curfval.j . . . . . . . . . . . 12 𝐽 = (Hom ‘𝐷)
3533, 34eqtr4di 2794 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
3635oveqd 7370 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐽𝑧))
3729fveq2d 6843 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (2nd𝑓) = (2nd𝐹))
3837oveqd 7370 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩))
3920fveq2d 6843 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = (Id‘𝐶))
40 curfval.1 . . . . . . . . . . . . 13 1 = (Id‘𝐶)
4139, 40eqtr4di 2794 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = 1 )
4241fveq1d 6841 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
43 eqidd 2737 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑔 = 𝑔)
4438, 42, 43oveq123d 7374 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔) = (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))
4536, 44mpteq12dv 5194 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))
4627, 27, 45mpoeq123dv 7428 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))))
4732, 46opeq12d 4836 . . . . . . 7 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩)
4823, 47mpteq12dv 5194 . . . . . 6 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
4920fveq2d 6843 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = (Hom ‘𝐶))
50 curfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
5149, 50eqtr4di 2794 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = 𝐻)
5251oveqd 7370 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
5337oveqd 7370 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩) = (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩))
5424fveq2d 6843 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = (Id‘𝐷))
55 curfval.i . . . . . . . . . . . 12 𝐼 = (Id‘𝐷)
5654, 55eqtr4di 2794 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = 𝐼)
5756fveq1d 6841 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑑)‘𝑧) = (𝐼𝑧))
5853, 43, 57oveq123d 7374 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)) = (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))
5927, 58mpteq12dv 5194 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧))) = (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))
6052, 59mpteq12dv 5194 . . . . . . 7 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))) = (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))
6123, 23, 60mpoeq123dv 7428 . . . . . 6 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧))))) = (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧))))))
6248, 61opeq12d 4836 . . . . 5 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩ = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
6313, 19, 62csbied2 3893 . . . 4 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩ = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
644, 12, 63csbied2 3893 . . 3 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) / 𝑐(2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩ = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
65 opex 5419 . . . 4 𝐶, 𝐷⟩ ∈ V
6665a1i 11 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ V)
67 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6867elexd 3463 . . 3 (𝜑𝐹 ∈ V)
69 opex 5419 . . . 4 ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩ ∈ V
7069a1i 11 . . 3 (𝜑 → ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩ ∈ V)
713, 64, 66, 68, 70ovmpod 7503 . 2 (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
721, 71eqtrid 2788 1 (𝜑𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  csb 3853  cop 4590  cmpt 5186  cfv 6493  (class class class)co 7353  cmpo 7355  1st c1st 7915  2nd c2nd 7916  Basecbs 17075  Hom chom 17136  Catccat 17536  Idccid 17537   Func cfunc 17732   ×c cxpc 18048   curryF ccurf 18091
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-curf 18095
This theorem is referenced by:  curf1fval  18105  curf2  18110  curfcl  18113  curfpropd  18114  curfuncf  18119
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