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Theorem prfcl 16771
 Description: The pairing of functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐷 is a functor ⟨𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfcl.p 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prfcl.t 𝑇 = (𝐷 ×c 𝐸)
prfcl.c (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfcl.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
Assertion
Ref Expression
prfcl (𝜑𝑃 ∈ (𝐶 Func 𝑇))

Proof of Theorem prfcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfcl.p . . . 4 𝑃 = (𝐹 ⟨,⟩F 𝐺)
2 eqid 2621 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2621 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
4 prfcl.c . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 prfcl.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
61, 2, 3, 4, 5prfval 16767 . . 3 (𝜑𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
7 fvex 6163 . . . . . . 7 (Base‘𝐶) ∈ V
87mptex 6446 . . . . . 6 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
97, 7mpt2ex 7199 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
108, 9op1std 7130 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
116, 10syl 17 . . . 4 (𝜑 → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
128, 9op2ndd 7131 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
136, 12syl 17 . . . 4 (𝜑 → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
1411, 13opeq12d 4383 . . 3 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
156, 14eqtr4d 2658 . 2 (𝜑𝑃 = ⟨(1st𝑃), (2nd𝑃)⟩)
16 prfcl.t . . . . 5 𝑇 = (𝐷 ×c 𝐸)
17 eqid 2621 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
18 eqid 2621 . . . . 5 (Base‘𝐸) = (Base‘𝐸)
1916, 17, 18xpcbas 16746 . . . 4 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
20 eqid 2621 . . . 4 (Hom ‘𝑇) = (Hom ‘𝑇)
21 eqid 2621 . . . 4 (Id‘𝐶) = (Id‘𝐶)
22 eqid 2621 . . . 4 (Id‘𝑇) = (Id‘𝑇)
23 eqid 2621 . . . 4 (comp‘𝐶) = (comp‘𝐶)
24 eqid 2621 . . . 4 (comp‘𝑇) = (comp‘𝑇)
25 funcrcl 16451 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
264, 25syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2726simpld 475 . . . 4 (𝜑𝐶 ∈ Cat)
2826simprd 479 . . . . 5 (𝜑𝐷 ∈ Cat)
29 funcrcl 16451 . . . . . . 7 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
305, 29syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
3130simprd 479 . . . . 5 (𝜑𝐸 ∈ Cat)
3216, 28, 31xpccat 16758 . . . 4 (𝜑𝑇 ∈ Cat)
33 relfunc 16450 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
34 1st2ndbr 7169 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3533, 4, 34sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
362, 17, 35funcf1 16454 . . . . . . . 8 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
3736ffvelrnda 6320 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
38 relfunc 16450 . . . . . . . . . 10 Rel (𝐶 Func 𝐸)
39 1st2ndbr 7169 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
4038, 5, 39sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
412, 18, 40funcf1 16454 . . . . . . . 8 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
4241ffvelrnda 6320 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
43 opelxpi 5113 . . . . . . 7 ((((1st𝐹)‘𝑥) ∈ (Base‘𝐷) ∧ ((1st𝐺)‘𝑥) ∈ (Base‘𝐸)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
4437, 42, 43syl2anc 692 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
45 eqid 2621 . . . . . 6 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
4644, 45fmptd 6346 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
4711feq1d 5992 . . . . 5 (𝜑 → ((1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)) ↔ (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸))))
4846, 47mpbird 247 . . . 4 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
49 eqid 2621 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
50 ovex 6638 . . . . . . 7 (𝑥(Hom ‘𝐶)𝑦) ∈ V
5150mptex 6446 . . . . . 6 ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V
5249, 51fnmpt2i 7191 . . . . 5 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))
5313fneq1d 5944 . . . . 5 (𝜑 → ((2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))))
5452, 53mpbiri 248 . . . 4 (𝜑 → (2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)))
55 eqid 2621 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
5635adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
57 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
58 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
592, 3, 55, 56, 57, 58funcf2 16456 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
6059ffvelrnda 6320 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
61 eqid 2621 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
6240adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
632, 3, 61, 62, 57, 58funcf2 16456 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
6463ffvelrnda 6320 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
65 opelxpi 5113 . . . . . . . 8 ((((𝑥(2nd𝐹)𝑦)‘) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ ((𝑥(2nd𝐺)𝑦)‘) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
6660, 64, 65syl2anc 692 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
674adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
685adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐶 Func 𝐸))
691, 2, 3, 67, 68, 57prf1 16768 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
701, 2, 3, 67, 68, 58prf1 16768 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
7169, 70oveq12d 6628 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩))
7237adantrr 752 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
7342adantrr 752 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
7436ffvelrnda 6320 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7574adantrl 751 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7641ffvelrnda 6320 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7776adantrl 751 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7816, 17, 18, 55, 61, 72, 73, 75, 77, 20xpchom2 16754 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
7971, 78eqtrd 2655 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
8079adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
8166, 80eleqtrrd 2701 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
82 eqid 2621 . . . . . 6 ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)
8381, 82fmptd 6346 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
8413oveqd 6627 . . . . . . 7 (𝜑 → (𝑥(2nd𝑃)𝑦) = (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦))
8549ovmpt4g 6743 . . . . . . . 8 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8651, 85mp3an3 1410 . . . . . . 7 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8784, 86sylan9eq 2675 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
8887feq1d 5992 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) ↔ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦))))
8983, 88mpbird 247 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
90 eqid 2621 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
9135adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
92 simpr 477 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
932, 21, 90, 91, 92funcid 16458 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
94 eqid 2621 . . . . . . 7 (Id‘𝐸) = (Id‘𝐸)
9540adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
962, 21, 94, 95, 92funcid 16458 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st𝐺)‘𝑥)))
9793, 96opeq12d 4383 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩ = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
984adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝐷))
995adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐸))
10027adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
1012, 3, 21, 100, 92catidcl 16271 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
1021, 2, 3, 98, 99, 92, 92, 101prf2 16770 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩)
1031, 2, 3, 98, 99, 92prf1 16768 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
104103fveq2d 6157 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
10528adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
10631adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
10716, 105, 106, 17, 18, 90, 94, 22, 37, 42xpcid 16757 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
108104, 107eqtrd 2655 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
10997, 102, 1083eqtr4d 2665 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑇)‘((1st𝑃)‘𝑥)))
110 eqid 2621 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
111353ad2ant1 1080 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
112 simp21 1092 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
113 simp22 1093 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
114 simp23 1094 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
115 simp3l 1087 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
116 simp3r 1088 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
1172, 3, 23, 110, 111, 112, 113, 114, 115, 116funcco 16459 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
118 eqid 2621 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
11953ad2ant1 1080 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐺 ∈ (𝐶 Func 𝐸))
12038, 119, 39sylancr 694 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
1212, 3, 23, 118, 120, 112, 113, 114, 115, 116funcco 16459 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓)))
122117, 121opeq12d 4383 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩ = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
12343ad2ant1 1080 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷))
124273ad2ant1 1080 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
1252, 3, 23, 124, 112, 113, 114, 115, 116catcocl 16274 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
1261, 2, 3, 123, 119, 112, 114, 125prf2 16770 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩)
1271, 2, 3, 123, 119, 112prf1 16768 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
1281, 2, 3, 123, 119, 113prf1 16768 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
129127, 128opeq12d 4383 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩ = ⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩)
1301, 2, 3, 123, 119, 114prf1 16768 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑧) = ⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)
131129, 130oveq12d 6628 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧)) = (⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩))
1321, 2, 3, 123, 119, 113, 114, 116prf2 16770 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝑃)𝑧)‘𝑔) = ⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩)
1331, 2, 3, 123, 119, 112, 113, 115prf2 16770 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑦)‘𝑓) = ⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩)
134131, 132, 133oveq123d 6631 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩))
135363ad2ant1 1080 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
136135, 112ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
137413ad2ant1 1080 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
138137, 112ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
139135, 113ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
140137, 113ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
141135, 114ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
142137, 114ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐸))
1432, 3, 55, 111, 112, 113funcf2 16456 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
144143, 115ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
1452, 3, 61, 120, 112, 113funcf2 16456 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
146145, 115ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
1472, 3, 55, 111, 113, 114funcf2 16456 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
148147, 116ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
1492, 3, 61, 120, 113, 114funcf2 16456 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
150149, 116ffvelrnd 6321 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑔) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
15116, 17, 18, 55, 61, 136, 138, 139, 140, 110, 118, 24, 141, 142, 144, 146, 148, 150xpcco2 16755 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
152134, 151eqtrd 2655 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
153122, 126, 1523eqtr4d 2665 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)))
1542, 19, 3, 20, 21, 22, 23, 24, 27, 32, 48, 54, 89, 109, 153isfuncd 16453 . . 3 (𝜑 → (1st𝑃)(𝐶 Func 𝑇)(2nd𝑃))
155 df-br 4619 . . 3 ((1st𝑃)(𝐶 Func 𝑇)(2nd𝑃) ↔ ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
156154, 155sylib 208 . 2 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
15715, 156eqeltrd 2698 1 (𝜑𝑃 ∈ (𝐶 Func 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  Vcvv 3189  ⟨cop 4159   class class class wbr 4618   ↦ cmpt 4678   × cxp 5077  Rel wrel 5084   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610   ↦ cmpt2 6612  1st c1st 7118  2nd c2nd 7119  Basecbs 15788  Hom chom 15880  compcco 15881  Catccat 16253  Idccid 16254   Func cfunc 16442   ×c cxpc 16736   ⟨,⟩F cprf 16739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-fz 12276  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-hom 15894  df-cco 15895  df-cat 16257  df-cid 16258  df-func 16446  df-xpc 16740  df-prf 16743 This theorem is referenced by:  prf1st  16772  prf2nd  16773  uncfcl  16803  uncf1  16804  uncf2  16805  yonedalem1  16840  yonedalem21  16841  yonedalem22  16846
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