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Theorem 2ndfcl 18254
Description: The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfcl.t 𝑇 = (𝐶 ×c 𝐷)
1stfcl.c (𝜑𝐶 ∈ Cat)
1stfcl.d (𝜑𝐷 ∈ Cat)
2ndfcl.p 𝑄 = (𝐶 2ndF 𝐷)
Assertion
Ref Expression
2ndfcl (𝜑𝑄 ∈ (𝑇 Func 𝐷))

Proof of Theorem 2ndfcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfcl.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 eqid 2735 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2735 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
41, 2, 3xpcbas 18234 . . . 4 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5 eqid 2735 . . . 4 (Hom ‘𝑇) = (Hom ‘𝑇)
6 1stfcl.c . . . 4 (𝜑𝐶 ∈ Cat)
7 1stfcl.d . . . 4 (𝜑𝐷 ∈ Cat)
8 2ndfcl.p . . . 4 𝑄 = (𝐶 2ndF 𝐷)
91, 4, 5, 6, 7, 82ndfval 18250 . . 3 (𝜑𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
10 fo2nd 8034 . . . . . . . 8 2nd :V–onto→V
11 fofun 6822 . . . . . . . 8 (2nd :V–onto→V → Fun 2nd )
1210, 11ax-mp 5 . . . . . . 7 Fun 2nd
13 fvex 6920 . . . . . . . 8 (Base‘𝐶) ∈ V
14 fvex 6920 . . . . . . . 8 (Base‘𝐷) ∈ V
1513, 14xpex 7772 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16 resfunexg 7235 . . . . . . 7 ((Fun 2nd ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
1712, 15, 16mp2an 692 . . . . . 6 (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
1815, 15mpoex 8103 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
1917, 18op2ndd 8024 . . . . 5 (𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩ → (2nd𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))))
209, 19syl 17 . . . 4 (𝜑 → (2nd𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))))
2120opeq2d 4885 . . 3 (𝜑 → ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑄)⟩ = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
229, 21eqtr4d 2778 . 2 (𝜑𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑄)⟩)
23 eqid 2735 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
24 eqid 2735 . . . 4 (Id‘𝑇) = (Id‘𝑇)
25 eqid 2735 . . . 4 (Id‘𝐷) = (Id‘𝐷)
26 eqid 2735 . . . 4 (comp‘𝑇) = (comp‘𝑇)
27 eqid 2735 . . . 4 (comp‘𝐷) = (comp‘𝐷)
281, 6, 7xpccat 18246 . . . 4 (𝜑𝑇 ∈ Cat)
29 f2ndres 8038 . . . . 5 (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷)
3029a1i 11 . . . 4 (𝜑 → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷))
31 eqid 2735 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
32 ovex 7464 . . . . . . 7 (𝑥(Hom ‘𝑇)𝑦) ∈ V
33 resfunexg 7235 . . . . . . 7 ((Fun 2nd ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
3412, 32, 33mp2an 692 . . . . . 6 (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
3531, 34fnmpoi 8094 . . . . 5 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))
3620fneq1d 6662 . . . . 5 (𝜑 → ((2nd𝑄) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))))
3735, 36mpbiri 258 . . . 4 (𝜑 → (2nd𝑄) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38 f2ndres 8038 . . . . . 6 (2nd ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((2nd𝑥)(Hom ‘𝐷)(2nd𝑦))
396adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
407adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
431, 4, 5, 39, 40, 8, 41, 422ndf2 18252 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
44 eqid 2735 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
451, 4, 44, 23, 5, 41, 42xpchom 18236 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(Hom ‘𝑇)𝑦) = (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦))))
4645reseq2d 6000 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) = (2nd ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))))
4743, 46eqtrd 2775 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑄)𝑦) = (2nd ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))))
4847feq1d 6721 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd𝑄)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)) ↔ (2nd ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((2nd𝑥)(Hom ‘𝐷)(2nd𝑦))))
4938, 48mpbiri 258 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑄)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))
50 fvres 6926 . . . . . . . 8 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd𝑥))
5150ad2antrl 728 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd𝑥))
52 fvres 6926 . . . . . . . 8 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd𝑦))
5352ad2antll 729 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd𝑦))
5451, 53oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) = ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))
5545, 54feq23d 6732 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) ↔ (𝑥(2nd𝑄)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((2nd𝑥)(Hom ‘𝐷)(2nd𝑦))))
5649, 55mpbird 257 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)))
5728adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat)
58 simpr 484 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
594, 5, 24, 57, 58catidcl 17727 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
6059fvresd 6927 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (2nd ‘((Id‘𝑇)‘𝑥)))
61 1st2nd2 8052 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6261adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6362fveq2d 6911 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1st𝑥), (2nd𝑥)⟩))
646adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
657adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
66 eqid 2735 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
67 xp1st 8045 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st𝑥) ∈ (Base‘𝐶))
6867adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st𝑥) ∈ (Base‘𝐶))
69 xp2nd 8046 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd𝑥) ∈ (Base‘𝐷))
7069adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd𝑥) ∈ (Base‘𝐷))
711, 64, 65, 2, 3, 66, 25, 24, 68, 70xpcid 18245 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
7263, 71eqtrd 2775 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
73 fvex 6920 . . . . . . . 8 ((Id‘𝐶)‘(1st𝑥)) ∈ V
74 fvex 6920 . . . . . . . 8 ((Id‘𝐷)‘(2nd𝑥)) ∈ V
7573, 74op2ndd 8024 . . . . . . 7 (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩ → (2nd ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd𝑥)))
7672, 75syl 17 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd𝑥)))
7760, 76eqtrd 2775 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd𝑥)))
781, 4, 5, 64, 65, 8, 58, 582ndf2 18252 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2nd𝑄)𝑥) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑥)))
7978fveq1d 6909 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
8050adantl 481 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd𝑥))
8180fveq2d 6911 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐷)‘((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐷)‘(2nd𝑥)))
8277, 79, 813eqtr4d 2785 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
83283ad2ant1 1132 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑇 ∈ Cat)
84 simp21 1205 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
85 simp22 1206 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
86 simp23 1207 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷)))
87 simp3l 1200 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦))
88 simp3r 1201 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))
894, 5, 26, 83, 84, 85, 86, 87, 88catcocl 17730 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧))
9089fvresd 6927 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (2nd ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
911, 4, 5, 26, 84, 85, 86, 87, 88, 27xpcco2nd 18241 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (2nd ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐷)(2nd𝑧))(2nd𝑓)))
9290, 91eqtrd 2775 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐷)(2nd𝑧))(2nd𝑓)))
9363ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐶 ∈ Cat)
9473ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐷 ∈ Cat)
951, 4, 5, 93, 94, 8, 84, 862ndf2 18252 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2nd𝑄)𝑧) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑧)))
9695fveq1d 6909 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd𝑄)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
9784fvresd 6927 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd𝑥))
9885fvresd 6927 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd𝑦))
9997, 98opeq12d 4886 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩ = ⟨(2nd𝑥), (2nd𝑦)⟩)
10086fvresd 6927 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (2nd𝑧))
10199, 100oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧)) = (⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐷)(2nd𝑧)))
1021, 4, 5, 93, 94, 8, 85, 862ndf2 18252 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑦(2nd𝑄)𝑧) = (2nd ↾ (𝑦(Hom ‘𝑇)𝑧)))
103102fveq1d 6909 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2nd𝑄)𝑧)‘𝑔) = ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔))
10488fvresd 6927 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (2nd𝑔))
105103, 104eqtrd 2775 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2nd𝑄)𝑧)‘𝑔) = (2nd𝑔))
1061, 4, 5, 93, 94, 8, 84, 852ndf2 18252 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2nd𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
107106fveq1d 6909 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd𝑄)𝑦)‘𝑓) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓))
10887fvresd 6927 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (2nd𝑓))
109107, 108eqtrd 2775 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd𝑄)𝑦)‘𝑓) = (2nd𝑓))
110101, 105, 109oveq123d 7452 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (((𝑦(2nd𝑄)𝑧)‘𝑔)(⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd𝑄)𝑦)‘𝑓)) = ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐷)(2nd𝑧))(2nd𝑓)))
11192, 96, 1103eqtr4d 2785 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd𝑄)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (((𝑦(2nd𝑄)𝑧)‘𝑔)(⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd𝑄)𝑦)‘𝑓)))
1124, 3, 5, 23, 24, 25, 26, 27, 28, 7, 30, 37, 56, 82, 111isfuncd 17916 . . 3 (𝜑 → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐷)(2nd𝑄))
113 df-br 5149 . . 3 ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐷)(2nd𝑄) ↔ ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑄)⟩ ∈ (𝑇 Func 𝐷))
114112, 113sylib 218 . 2 (𝜑 → ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑄)⟩ ∈ (𝑇 Func 𝐷))
11522, 114eqeltrd 2839 1 (𝜑𝑄 ∈ (𝑇 Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148   × cxp 5687  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710   Func cfunc 17905   ×c cxpc 18224   2ndF c2ndf 18226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-func 17909  df-xpc 18228  df-2ndf 18230
This theorem is referenced by:  prf2nd  18261  1st2ndprf  18262  uncfcl  18292  uncf1  18293  uncf2  18294  curf2ndf  18304  yonedalem1  18329  yonedalem21  18330  yonedalem22  18335
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