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Theorem isfunc 16725
Description: Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
isfunc.b 𝐵 = (Base‘𝐷)
isfunc.c 𝐶 = (Base‘𝐸)
isfunc.h 𝐻 = (Hom ‘𝐷)
isfunc.j 𝐽 = (Hom ‘𝐸)
isfunc.1 1 = (Id‘𝐷)
isfunc.i 𝐼 = (Id‘𝐸)
isfunc.x · = (comp‘𝐷)
isfunc.o 𝑂 = (comp‘𝐸)
isfunc.d (𝜑𝐷 ∈ Cat)
isfunc.e (𝜑𝐸 ∈ Cat)
Assertion
Ref Expression
isfunc (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
Distinct variable groups:   𝑚,𝑛,𝑥,𝑦,𝑧,𝐵   𝐷,𝑚,𝑛,𝑥,𝑦,𝑧   𝑚,𝐸,𝑛,𝑥,𝑦,𝑧   𝑚,𝐻,𝑛,𝑥,𝑦,𝑧   𝑚,𝐹,𝑛,𝑥,𝑦,𝑧   𝑚,𝐺,𝑛,𝑥,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝜑,𝑚,𝑛,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑚,𝑛)   · (𝑥,𝑦,𝑧,𝑚,𝑛)   1 (𝑥,𝑦,𝑧,𝑚,𝑛)   𝐼(𝑥,𝑦,𝑧,𝑚,𝑛)   𝐽(𝑚,𝑛)   𝑂(𝑥,𝑦,𝑧,𝑚,𝑛)

Proof of Theorem isfunc
Dummy variables 𝑏 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfunc.d . . . 4 (𝜑𝐷 ∈ Cat)
2 isfunc.e . . . 4 (𝜑𝐸 ∈ Cat)
3 fvexd 6364 . . . . . . 7 ((𝑑 = 𝐷𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
4 simpl 474 . . . . . . . . 9 ((𝑑 = 𝐷𝑒 = 𝐸) → 𝑑 = 𝐷)
54fveq2d 6356 . . . . . . . 8 ((𝑑 = 𝐷𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
6 isfunc.b . . . . . . . 8 𝐵 = (Base‘𝐷)
75, 6syl6eqr 2812 . . . . . . 7 ((𝑑 = 𝐷𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
8 simpr 479 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
9 simplr 809 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
109fveq2d 6356 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
11 isfunc.c . . . . . . . . . . . 12 𝐶 = (Base‘𝐸)
1210, 11syl6eqr 2812 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
138, 12feq23d 6201 . . . . . . . . . 10 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓:𝐵𝐶))
14 fvex 6362 . . . . . . . . . . . 12 (Base‘𝐸) ∈ V
1511, 14eqeltri 2835 . . . . . . . . . . 11 𝐶 ∈ V
16 fvex 6362 . . . . . . . . . . . 12 (Base‘𝐷) ∈ V
176, 16eqeltri 2835 . . . . . . . . . . 11 𝐵 ∈ V
1815, 17elmap 8052 . . . . . . . . . 10 (𝑓 ∈ (𝐶𝑚 𝐵) ↔ 𝑓:𝐵𝐶)
1913, 18syl6bbr 278 . . . . . . . . 9 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑓:𝑏⟶(Base‘𝑒) ↔ 𝑓 ∈ (𝐶𝑚 𝐵)))
208sqxpeqd 5298 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
2120ixpeq1d 8086 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)))
229fveq2d 6356 . . . . . . . . . . . . . . 15 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = (Hom ‘𝐸))
23 isfunc.j . . . . . . . . . . . . . . 15 𝐽 = (Hom ‘𝐸)
2422, 23syl6eqr 2812 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑒) = 𝐽)
2524oveqd 6830 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) = ((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))))
26 simpll 807 . . . . . . . . . . . . . . . 16 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
2726fveq2d 6356 . . . . . . . . . . . . . . 15 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = (Hom ‘𝐷))
28 isfunc.h . . . . . . . . . . . . . . 15 𝐻 = (Hom ‘𝐷)
2927, 28syl6eqr 2812 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Hom ‘𝑑) = 𝐻)
3029fveq1d 6354 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑑)‘𝑧) = (𝐻𝑧))
3125, 30oveq12d 6831 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = (((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
3231ixpeq2dv 8090 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
3321, 32eqtrd 2794 . . . . . . . . . 10 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
3433eleq2d 2825 . . . . . . . . 9 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ↔ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))))
3526fveq2d 6356 . . . . . . . . . . . . . . 15 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = (Id‘𝐷))
36 isfunc.1 . . . . . . . . . . . . . . 15 1 = (Id‘𝐷)
3735, 36syl6eqr 2812 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑑) = 1 )
3837fveq1d 6354 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑑)‘𝑥) = ( 1𝑥))
3938fveq2d 6356 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((𝑥𝑔𝑥)‘( 1𝑥)))
409fveq2d 6356 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = (Id‘𝐸))
41 isfunc.i . . . . . . . . . . . . . 14 𝐼 = (Id‘𝐸)
4240, 41syl6eqr 2812 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Id‘𝑒) = 𝐼)
4342fveq1d 6354 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((Id‘𝑒)‘(𝑓𝑥)) = (𝐼‘(𝑓𝑥)))
4439, 43eqeq12d 2775 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ↔ ((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥))))
4529oveqd 6830 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑥(Hom ‘𝑑)𝑦) = (𝑥𝐻𝑦))
4629oveqd 6830 . . . . . . . . . . . . . . 15 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑦(Hom ‘𝑑)𝑧) = (𝑦𝐻𝑧))
4726fveq2d 6356 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = (comp‘𝐷))
48 isfunc.x . . . . . . . . . . . . . . . . . . . 20 · = (comp‘𝐷)
4947, 48syl6eqr 2812 . . . . . . . . . . . . . . . . . . 19 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑑) = · )
5049oveqd 6830 . . . . . . . . . . . . . . . . . 18 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧) = (⟨𝑥, 𝑦· 𝑧))
5150oveqd 6830 . . . . . . . . . . . . . . . . 17 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚) = (𝑛(⟨𝑥, 𝑦· 𝑧)𝑚))
5251fveq2d 6356 . . . . . . . . . . . . . . . 16 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)))
539fveq2d 6356 . . . . . . . . . . . . . . . . . . 19 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = (comp‘𝐸))
54 isfunc.o . . . . . . . . . . . . . . . . . . 19 𝑂 = (comp‘𝐸)
5553, 54syl6eqr 2812 . . . . . . . . . . . . . . . . . 18 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (comp‘𝑒) = 𝑂)
5655oveqd 6830 . . . . . . . . . . . . . . . . 17 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧)) = (⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧)))
5756oveqd 6830 . . . . . . . . . . . . . . . 16 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))
5852, 57eqeq12d 2775 . . . . . . . . . . . . . . 15 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))
5946, 58raleqbidv 3291 . . . . . . . . . . . . . 14 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))
6045, 59raleqbidv 3291 . . . . . . . . . . . . 13 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))
618, 60raleqbidv 3291 . . . . . . . . . . . 12 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))
628, 61raleqbidv 3291 . . . . . . . . . . 11 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))
6344, 62anbi12d 749 . . . . . . . . . 10 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))))
648, 63raleqbidv 3291 . . . . . . . . 9 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))))
6519, 34, 643anbi123d 1548 . . . . . . . 8 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))))
66 df-3an 1074 . . . . . . . 8 ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))))
6765, 66syl6bb 276 . . . . . . 7 (((𝑑 = 𝐷𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ((𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))))
683, 7, 67sbcied2 3614 . . . . . 6 ((𝑑 = 𝐷𝑒 = 𝐸) → ([(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))))
6968opabbidv 4868 . . . . 5 ((𝑑 = 𝐷𝑒 = 𝐸) → {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
70 df-func 16719 . . . . 5 Func = (𝑑 ∈ Cat, 𝑒 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑑) / 𝑏](𝑓:𝑏⟶(Base‘𝑒) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑒)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑑)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑑)‘𝑥)) = ((Id‘𝑒)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑑)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑑)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑑)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑒)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
71 ovex 6841 . . . . . . 7 (𝐶𝑚 𝐵) ∈ V
72 snex 5057 . . . . . . . 8 {𝑓} ∈ V
73 ovex 6841 . . . . . . . . . 10 (((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∈ V
7473rgenw 3062 . . . . . . . . 9 𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∈ V
75 ixpexg 8098 . . . . . . . . 9 (∀𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∈ V → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∈ V)
7674, 75ax-mp 5 . . . . . . . 8 X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∈ V
7772, 76xpex 7127 . . . . . . 7 ({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∈ V
7871, 77iunex 7312 . . . . . 6 𝑓 ∈ (𝐶𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∈ V
79 simpl 474 . . . . . . . . . 10 (((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) → (𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))))
8079anim2i 594 . . . . . . . . 9 ((𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))) → (𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))))
81802eximi 1912 . . . . . . . 8 (∃𝑓𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))) → ∃𝑓𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))))
82 elopab 5133 . . . . . . . 8 (𝑑 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} ↔ ∃𝑓𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))))
83 eliunxp 5415 . . . . . . . 8 (𝑑 𝑓 ∈ (𝐶𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ↔ ∃𝑓𝑔(𝑑 = ⟨𝑓, 𝑔⟩ ∧ (𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))))
8481, 82, 833imtr4i 281 . . . . . . 7 (𝑑 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} → 𝑑 𝑓 ∈ (𝐶𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))))
8584ssriv 3748 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} ⊆ 𝑓 ∈ (𝐶𝑚 𝐵)({𝑓} × X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
8678, 85ssexi 4955 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} ∈ V
8769, 70, 86ovmpt2a 6956 . . . 4 ((𝐷 ∈ Cat ∧ 𝐸 ∈ Cat) → (𝐷 Func 𝐸) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
881, 2, 87syl2anc 696 . . 3 (𝜑 → (𝐷 Func 𝐸) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
8988breqd 4815 . 2 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺))
90 brabv 6864 . . . 4 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
91 elex 3352 . . . . . 6 (𝐹 ∈ (𝐶𝑚 𝐵) → 𝐹 ∈ V)
92 elex 3352 . . . . . 6 (𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) → 𝐺 ∈ V)
9391, 92anim12i 591 . . . . 5 ((𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
94933adant3 1127 . . . 4 ((𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
95 simpl 474 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
9695eleq1d 2824 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ∈ (𝐶𝑚 𝐵) ↔ 𝐹 ∈ (𝐶𝑚 𝐵)))
97 simpr 479 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
9895fveq1d 6354 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘(1st𝑧)) = (𝐹‘(1st𝑧)))
9995fveq1d 6354 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘(2nd𝑧)) = (𝐹‘(2nd𝑧)))
10098, 99oveq12d 6831 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) = ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))))
101100oveq1d 6828 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
102101ixpeq2dv 8090 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
10397, 102eleq12d 2833 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))))
10497oveqd 6830 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑥) = (𝑥𝐺𝑥))
105104fveq1d 6354 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑥𝑔𝑥)‘( 1𝑥)) = ((𝑥𝐺𝑥)‘( 1𝑥)))
10695fveq1d 6354 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
107106fveq2d 6356 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝐼‘(𝑓𝑥)) = (𝐼‘(𝐹𝑥)))
108105, 107eqeq12d 2775 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ↔ ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥))))
10997oveqd 6830 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑧) = (𝑥𝐺𝑧))
110109fveq1d 6354 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)))
11195fveq1d 6354 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑦) = (𝐹𝑦))
112106, 111opeq12d 4561 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑔 = 𝐺) → ⟨(𝑓𝑥), (𝑓𝑦)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
11395fveq1d 6354 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑧) = (𝐹𝑧))
114112, 113oveq12d 6831 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑔 = 𝐺) → (⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧)))
11597oveqd 6830 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
116115fveq1d 6354 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑦𝑔𝑧)‘𝑛) = ((𝑦𝐺𝑧)‘𝑛))
11797oveqd 6830 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
118117fveq1d 6354 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑥𝑔𝑦)‘𝑚) = ((𝑥𝐺𝑦)‘𝑚))
119114, 116, 118oveq123d 6834 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
120110, 119eqeq12d 2775 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
1211202ralbidv 3127 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
1221212ralbidv 3127 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)) ↔ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
123108, 122anbi12d 749 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
124123ralbidv 3124 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))) ↔ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
12596, 103, 1243anbi123d 1548 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
12666, 125syl5bbr 274 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚)))) ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
127 eqid 2760 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}
128126, 127brabga 5139 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
12990, 94, 128pm5.21nii 367 . . 3 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
13015, 17elmap 8052 . . . 4 (𝐹 ∈ (𝐶𝑚 𝐵) ↔ 𝐹:𝐵𝐶)
1311303anbi1i 1161 . . 3 ((𝐹 ∈ (𝐶𝑚 𝐵) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))) ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
132129, 131bitri 264 . 2 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐶𝑚 𝐵) ∧ 𝑔X𝑧 ∈ (𝐵 × 𝐵)(((𝑓‘(1st𝑧))𝐽(𝑓‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))) ∧ ∀𝑥𝐵 (((𝑥𝑔𝑥)‘( 1𝑥)) = (𝐼‘(𝑓𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩𝑂(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))}𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
13389, 132syl6bb 276 1 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wral 3050  Vcvv 3340  [wsbc 3576  {csn 4321  cop 4327   ciun 4672   class class class wbr 4804  {copab 4864   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  Xcixp 8074  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  Idccid 16527   Func cfunc 16715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-ixp 8075  df-func 16719
This theorem is referenced by:  isfuncd  16726  funcf1  16727  funcixp  16728  funcid  16731  funcco  16732  idfucl  16742  cofucl  16749  funcres2b  16758  funcpropd  16761
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