MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curfval Structured version   Visualization version   GIF version

Theorem curfval 18222
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 18213 . . . 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 6917 . . . 4 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) ∈ V)
5 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → 𝑒 = ⟨𝐶, 𝐷⟩)
65fveq2d 6906 . . . . 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 8011 . . . . . . 7 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
107, 8, 9syl2anc 582 . . . . . 6 (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
1110adantr 479 . . . . 5 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
126, 11eqtrd 2768 . . . 4 ((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) → (1st𝑒) = 𝐶)
13 fvexd 6917 . . . . 5 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) ∈ V)
145adantr 479 . . . . . . 7 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → 𝑒 = ⟨𝐶, 𝐷⟩)
1514fveq2d 6906 . . . . . 6 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd𝑒) = (2nd ‘⟨𝐶, 𝐷⟩))
16 op2ndg 8012 . . . . . . . 8 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
177, 8, 16syl2anc 582 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
1817ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
1915, 18eqtrd 2768 . . . . 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 6906 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
22 curfval.a . . . . . . . 8 𝐴 = (Base‘𝐶)
2321, 22eqtr4di 2786 . . . . . . 7 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐴)
24 simpr 483 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
2524fveq2d 6906 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Base‘𝑑) = (Base‘𝐷))
26 curfval.b . . . . . . . . . 10 𝐵 = (Base‘𝐷)
2725, 26eqtr4di 2786 . . . . . . . . 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 6906 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (1st𝑓) = (1st𝐹))
3130oveqd 7443 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(1st𝑓)𝑦) = (𝑥(1st𝐹)𝑦))
3227, 31mpteq12dv 5243 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)) = (𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)))
3324fveq2d 6906 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
34 curfval.j . . . . . . . . . . . 12 𝐽 = (Hom ‘𝐷)
3533, 34eqtr4di 2786 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
3635oveqd 7443 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐽𝑧))
3729fveq2d 6906 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (2nd𝑓) = (2nd𝐹))
3837oveqd 7443 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩) = (⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩))
3920fveq2d 6906 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = (Id‘𝐶))
40 curfval.1 . . . . . . . . . . . . 13 1 = (Id‘𝐶)
4139, 40eqtr4di 2786 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑐) = 1 )
4241fveq1d 6904 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
43 eqidd 2729 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑔 = 𝑔)
4438, 42, 43oveq123d 7447 . . . . . . . . . 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 5243 . . . . . . . . 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 7501 . . . . . . . 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 4886 . . . . . . 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 5243 . . . . . 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 6906 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = (Hom ‘𝐶))
50 curfval.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
5149, 50eqtr4di 2786 . . . . . . . . 9 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Hom ‘𝑐) = 𝐻)
5251oveqd 7443 . . . . . . . 8 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
5337oveqd 7443 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩) = (⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩))
5424fveq2d 6906 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = (Id‘𝐷))
55 curfval.i . . . . . . . . . . . 12 𝐼 = (Id‘𝐷)
5654, 55eqtr4di 2786 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (Id‘𝑑) = 𝐼)
5756fveq1d 6904 . . . . . . . . . 10 ((((𝜑 ∧ (𝑒 = ⟨𝐶, 𝐷⟩ ∧ 𝑓 = 𝐹)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((Id‘𝑑)‘𝑧) = (𝐼𝑧))
5853, 43, 57oveq123d 7447 . . . . . . . . 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 5243 . . . . . . . 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 5243 . . . . . . 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 7501 . . . . . 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 4886 . . . . 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 3934 . . . 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 3934 . . 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 5470 . . . 4 𝐶, 𝐷⟩ ∈ V
6665a1i 11 . . 3 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ V)
67 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6867elexd 3494 . . 3 (𝜑𝐹 ∈ V)
69 opex 5470 . . . 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 7579 . 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 2780 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 394   = wceq 1533  wcel 2098  Vcvv 3473  csb 3894  cop 4638  cmpt 5235  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7997  2nd c2nd 7998  Basecbs 17187  Hom chom 17251  Catccat 17651  Idccid 17652   Func cfunc 17847   ×c cxpc 18166   curryF ccurf 18209
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-curf 18213
This theorem is referenced by:  curf1fval  18223  curf2  18228  curfcl  18231  curfpropd  18232  curfuncf  18237
  Copyright terms: Public domain W3C validator