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Theorem 1stfcl 18242
Description: The first projection functor is a functor onto the left 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)
1stfcl.p 𝑃 = (𝐶 1stF 𝐷)
Assertion
Ref Expression
1stfcl (𝜑𝑃 ∈ (𝑇 Func 𝐶))

Proof of Theorem 1stfcl
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 2737 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2737 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
41, 2, 3xpcbas 18223 . . . 4 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5 eqid 2737 . . . 4 (Hom ‘𝑇) = (Hom ‘𝑇)
6 1stfcl.c . . . 4 (𝜑𝐶 ∈ Cat)
7 1stfcl.d . . . 4 (𝜑𝐷 ∈ Cat)
8 1stfcl.p . . . 4 𝑃 = (𝐶 1stF 𝐷)
91, 4, 5, 6, 7, 81stfval 18236 . . 3 (𝜑𝑃 = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
10 fo1st 8034 . . . . . . . 8 1st :V–onto→V
11 fofun 6821 . . . . . . . 8 (1st :V–onto→V → Fun 1st )
1210, 11ax-mp 5 . . . . . . 7 Fun 1st
13 fvex 6919 . . . . . . . 8 (Base‘𝐶) ∈ V
14 fvex 6919 . . . . . . . 8 (Base‘𝐷) ∈ V
1513, 14xpex 7773 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16 resfunexg 7235 . . . . . . 7 ((Fun 1st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
1712, 15, 16mp2an 692 . . . . . 6 (1st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
1815, 15mpoex 8104 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
1917, 18op2ndd 8025 . . . . 5 (𝑃 = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩ → (2nd𝑃) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))))
209, 19syl 17 . . . 4 (𝜑 → (2nd𝑃) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))))
2120opeq2d 4880 . . 3 (𝜑 → ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩ = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
229, 21eqtr4d 2780 . 2 (𝜑𝑃 = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩)
23 eqid 2737 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
24 eqid 2737 . . . 4 (Id‘𝑇) = (Id‘𝑇)
25 eqid 2737 . . . 4 (Id‘𝐶) = (Id‘𝐶)
26 eqid 2737 . . . 4 (comp‘𝑇) = (comp‘𝑇)
27 eqid 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
281, 6, 7xpccat 18235 . . . 4 (𝜑𝑇 ∈ Cat)
29 f1stres 8038 . . . . 5 (1st ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐶)
3029a1i 11 . . . 4 (𝜑 → (1st ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐶))
31 eqid 2737 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))
32 ovex 7464 . . . . . . 7 (𝑥(Hom ‘𝑇)𝑦) ∈ V
33 resfunexg 7235 . . . . . . 7 ((Fun 1st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
3412, 32, 33mp2an 692 . . . . . 6 (1st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
3531, 34fnmpoi 8095 . . . . 5 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))
3620fneq1d 6661 . . . . 5 (𝜑 → ((2nd𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))))
3735, 36mpbiri 258 . . . 4 (𝜑 → (2nd𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38 f1stres 8038 . . . . . 6 (1st ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦))
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, 421stf2 18238 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑃)𝑦) = (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))
44 eqid 2737 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
451, 4, 23, 44, 5, 41, 42xpchom 18225 . . . . . . . . 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 5997 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (1st ↾ (𝑥(Hom ‘𝑇)𝑦)) = (1st ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))))
4743, 46eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑃)𝑦) = (1st ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))))
4847feq1d 6720 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd𝑃)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦)) ↔ (1st ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦))))
4938, 48mpbiri 258 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑃)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦)))
50 fvres 6925 . . . . . . . 8 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
5150ad2antrl 728 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
52 fvres 6925 . . . . . . . 8 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1st𝑦))
5352ad2antll 729 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1st𝑦))
5451, 53oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) = ((1st𝑥)(Hom ‘𝐶)(1st𝑦)))
5545, 54feq23d 6731 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) ↔ (𝑥(2nd𝑃)𝑦):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦))))
5649, 55mpbird 257 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1st ↾ ((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 17725 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
6059fvresd 6926 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1st ‘((Id‘𝑇)‘𝑥)))
61 1st2nd2 8053 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6261adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6362fveq2d 6910 . . . . . . . 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 2737 . . . . . . . . 9 (Id‘𝐷) = (Id‘𝐷)
67 xp1st 8046 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st𝑥) ∈ (Base‘𝐶))
6867adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st𝑥) ∈ (Base‘𝐶))
69 xp2nd 8047 . . . . . . . . . 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, 25, 66, 24, 68, 70xpcid 18234 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
7263, 71eqtrd 2777 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
73 fvex 6919 . . . . . . . 8 ((Id‘𝐶)‘(1st𝑥)) ∈ V
74 fvex 6919 . . . . . . . 8 ((Id‘𝐷)‘(2nd𝑥)) ∈ V
7573, 74op1std 8024 . . . . . . 7 (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩ → (1st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
7672, 75syl 17 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
7760, 76eqtrd 2777 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
781, 4, 5, 64, 65, 8, 58, 581stf2 18238 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2nd𝑃)𝑥) = (1st ↾ (𝑥(Hom ‘𝑇)𝑥)))
7978fveq1d 6908 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
8050adantl 481 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
8180fveq2d 6910 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐶)‘((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
8277, 79, 813eqtr4d 2787 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
83283ad2ant1 1134 . . . . . . . 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 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))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
85 simp22 1208 . . . . . . . 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 1209 . . . . . . . 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 1202 . . . . . . . 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 1203 . . . . . . . 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 17728 . . . . . . 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 6926 . . . . . 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))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (1st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
911, 4, 5, 26, 84, 85, 86, 87, 88, 27xpcco1st 18229 . . . . . 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))) → (1st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝐶)(1st𝑧))(1st𝑓)))
9290, 91eqtrd 2777 . . . . 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))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝐶)(1st𝑧))(1st𝑓)))
9363ad2ant1 1134 . . . . . . 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 1134 . . . . . . 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, 861stf2 18238 . . . . . 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(2nd𝑃)𝑧) = (1st ↾ (𝑥(Hom ‘𝑇)𝑧)))
9695fveq1d 6908 . . . . 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(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
9784fvresd 6926 . . . . . . . 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))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
9885fvresd 6926 . . . . . . . 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))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1st𝑦))
9997, 98opeq12d 4881 . . . . . . 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))) → ⟨((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩ = ⟨(1st𝑥), (1st𝑦)⟩)
10086fvresd 6926 . . . . . . 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))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (1st𝑧))
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))) → (⟨((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧)) = (⟨(1st𝑥), (1st𝑦)⟩(comp‘𝐶)(1st𝑧)))
1021, 4, 5, 93, 94, 8, 85, 861stf2 18238 . . . . . . . 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(2nd𝑃)𝑧) = (1st ↾ (𝑦(Hom ‘𝑇)𝑧)))
103102fveq1d 6908 . . . . . . 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(2nd𝑃)𝑧)‘𝑔) = ((1st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔))
10488fvresd 6926 . . . . . . 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))) → ((1st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (1st𝑔))
105103, 104eqtrd 2777 . . . . . 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(2nd𝑃)𝑧)‘𝑔) = (1st𝑔))
1061, 4, 5, 93, 94, 8, 84, 851stf2 18238 . . . . . . . 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(2nd𝑃)𝑦) = (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))
107106fveq1d 6908 . . . . . . 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(2nd𝑃)𝑦)‘𝑓) = ((1st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓))
10887fvresd 6926 . . . . . . 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))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (1st𝑓))
109107, 108eqtrd 2777 . . . . . 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(2nd𝑃)𝑦)‘𝑓) = (1st𝑓))
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(2nd𝑃)𝑧)‘𝑔)(⟨((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝐶)(1st𝑧))(1st𝑓)))
11192, 96, 1103eqtr4d 2787 . . . 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(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)))
1124, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 82, 111isfuncd 17910 . . 3 (𝜑 → (1st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2nd𝑃))
113 df-br 5144 . . 3 ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2nd𝑃) ↔ ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩ ∈ (𝑇 Func 𝐶))
114112, 113sylib 218 . 2 (𝜑 → ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩ ∈ (𝑇 Func 𝐶))
11522, 114eqeltrd 2841 1 (𝜑𝑃 ∈ (𝑇 Func 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143   × cxp 5683  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708   Func cfunc 17899   ×c cxpc 18213   1stF c1stf 18214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-func 17903  df-xpc 18217  df-1stf 18218
This theorem is referenced by:  prf1st  18249  1st2ndprf  18251  uncfcl  18280  uncf1  18281  uncf2  18282  diagcl  18286  diag11  18288  diag12  18289  diag2  18290  yonedalem1  18317  yonedalem21  18318  yonedalem22  18323
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