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Theorem prfcl 16771
Description: The pairing of functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐷 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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