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Theorem 1stfcl 17036
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 2758 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2758 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
41, 2, 3xpcbas 17017 . . . 4 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5 eqid 2758 . . . 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 17030 . . 3 (𝜑𝑃 = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
10 fo1st 7351 . . . . . . . 8 1st :V–onto→V
11 fofun 6275 . . . . . . . 8 (1st :V–onto→V → Fun 1st )
1210, 11ax-mp 5 . . . . . . 7 Fun 1st
13 fvex 6360 . . . . . . . 8 (Base‘𝐶) ∈ V
14 fvex 6360 . . . . . . . 8 (Base‘𝐷) ∈ V
1513, 14xpex 7125 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16 resfunexg 6641 . . . . . . 7 ((Fun 1st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
1712, 15, 16mp2an 710 . . . . . 6 (1st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
1815, 15mpt2ex 7413 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
1917, 18op2ndd 7342 . . . . 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 4558 . . 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 2795 . 2 (𝜑𝑃 = ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩)
23 eqid 2758 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
24 eqid 2758 . . . 4 (Id‘𝑇) = (Id‘𝑇)
25 eqid 2758 . . . 4 (Id‘𝐶) = (Id‘𝐶)
26 eqid 2758 . . . 4 (comp‘𝑇) = (comp‘𝑇)
27 eqid 2758 . . . 4 (comp‘𝐶) = (comp‘𝐶)
281, 6, 7xpccat 17029 . . . 4 (𝜑𝑇 ∈ Cat)
29 f1stres 7355 . . . . 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 2758 . . . . . 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 6839 . . . . . . 7 (𝑥(Hom ‘𝑇)𝑦) ∈ V
33 resfunexg 6641 . . . . . . 7 ((Fun 1st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
3412, 32, 33mp2an 710 . . . . . 6 (1st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
3531, 34fnmpt2i 7405 . . . . 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 6140 . . . . 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 248 . . . 4 (𝜑 → (2nd𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38 f1stres 7355 . . . . . 6 (1st ↾ (((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))):(((1st𝑥)(Hom ‘𝐶)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐷)(2nd𝑦)))⟶((1st𝑥)(Hom ‘𝐶)(1st𝑦))
396adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
407adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41 simprl 811 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42 simprr 813 . . . . . . . . 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 17032 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd𝑃)𝑦) = (1st ↾ (𝑥(Hom ‘𝑇)𝑦)))
44 eqid 2758 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
451, 4, 23, 44, 5, 41, 42xpchom 17019 . . . . . . . . 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 5549 . . . . . . . 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 2792 . . . . . . 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 6189 . . . . . 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 248 . . . . 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 6366 . . . . . . . 8 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
5150ad2antrl 766 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
52 fvres 6366 . . . . . . . 8 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1st𝑦))
5352ad2antll 767 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1st𝑦))
5451, 53oveq12d 6829 . . . . . 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 6199 . . . . 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 247 . . . 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 472 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat)
58 simpr 479 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
594, 5, 24, 57, 58catidcl 16542 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60 fvres 6366 . . . . . . 7 (((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1st ‘((Id‘𝑇)‘𝑥)))
6159, 60syl 17 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1st ‘((Id‘𝑇)‘𝑥)))
62 1st2nd2 7370 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6362adantl 473 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6463fveq2d 6354 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1st𝑥), (2nd𝑥)⟩))
656adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
667adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
67 eqid 2758 . . . . . . . . 9 (Id‘𝐷) = (Id‘𝐷)
68 xp1st 7363 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st𝑥) ∈ (Base‘𝐶))
6968adantl 473 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st𝑥) ∈ (Base‘𝐶))
70 xp2nd 7364 . . . . . . . . . 10 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd𝑥) ∈ (Base‘𝐷))
7170adantl 473 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd𝑥) ∈ (Base‘𝐷))
721, 65, 66, 2, 3, 25, 67, 24, 69, 71xpcid 17028 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
7364, 72eqtrd 2792 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩)
74 fvex 6360 . . . . . . . 8 ((Id‘𝐶)‘(1st𝑥)) ∈ V
75 fvex 6360 . . . . . . . 8 ((Id‘𝐷)‘(2nd𝑥)) ∈ V
7674, 75op1std 7341 . . . . . . 7 (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐷)‘(2nd𝑥))⟩ → (1st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
7773, 76syl 17 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
7861, 77eqtrd 2792 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
791, 4, 5, 65, 66, 8, 58, 581stf2 17032 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2nd𝑃)𝑥) = (1st ↾ (𝑥(Hom ‘𝑇)𝑥)))
8079fveq1d 6352 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((1st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
8150adantl 473 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1st𝑥))
8281fveq2d 6354 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐶)‘((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
8378, 80, 823eqtr4d 2802 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
84283ad2ant1 1128 . . . . . . . 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)
85 simp21 1249 . . . . . . . 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)))
86 simp22 1250 . . . . . . . 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)))
87 simp23 1251 . . . . . . . 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)))
88 simp3l 1244 . . . . . . . 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))
89 simp3r 1245 . . . . . . . 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))
904, 5, 26, 84, 85, 86, 87, 88, 89catcocl 16545 . . . . . . 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))
91 fvres 6366 . . . . . . 7 ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (1st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
9290, 91syl 17 . . . . . 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)))
931, 4, 5, 26, 85, 86, 87, 88, 89, 27xpcco1st 17023 . . . . . 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)))
9492, 93eqtrd 2792 . . . . 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)))
9563ad2ant1 1128 . . . . . . 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)
9673ad2ant1 1128 . . . . . . 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)
971, 4, 5, 95, 96, 8, 85, 871stf2 17032 . . . . . 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)))
9897fveq1d 6352 . . . . 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)))
9985, 50syl 17 . . . . . . . 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))
10086, 52syl 17 . . . . . . . 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))
10199, 100opeq12d 4559 . . . . . . 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)⟩)
102 fvres 6366 . . . . . . . 8 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (1st𝑧))
10387, 102syl 17 . . . . . . 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))
104101, 103oveq12d 6829 . . . . . 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)))
1051, 4, 5, 95, 96, 8, 86, 871stf2 17032 . . . . . . . 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)))
106105fveq1d 6352 . . . . . . 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))
107 fvres 6366 . . . . . . . 8 (𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧) → ((1st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (1st𝑔))
10889, 107syl 17 . . . . . . 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))
109106, 108eqtrd 2792 . . . . . 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))
1101, 4, 5, 95, 96, 8, 85, 861stf2 17032 . . . . . . . 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)))
111110fveq1d 6352 . . . . . . 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))
112 fvres 6366 . . . . . . . 8 (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) → ((1st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (1st𝑓))
11388, 112syl 17 . . . . . . 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))
114111, 113eqtrd 2792 . . . . . 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))
115104, 109, 114oveq123d 6832 . . . . 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)))
11694, 98, 1153eqtr4d 2802 . . . 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)))
1174, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 83, 116isfuncd 16724 . . 3 (𝜑 → (1st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2nd𝑃))
118 df-br 4803 . . 3 ((1st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2nd𝑃) ↔ ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩ ∈ (𝑇 Func 𝐶))
119117, 118sylib 208 . 2 (𝜑 → ⟨(1st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd𝑃)⟩ ∈ (𝑇 Func 𝐶))
12022, 119eqeltrd 2837 1 (𝜑𝑃 ∈ (𝑇 Func 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  Vcvv 3338  cop 4325   class class class wbr 4802   × cxp 5262  cres 5266  Fun wfun 6041   Fn wfn 6042  wf 6043  ontowfo 6045  cfv 6047  (class class class)co 6811  cmpt2 6813  1st c1st 7329  2nd c2nd 7330  Basecbs 16057  Hom chom 16152  compcco 16153  Catccat 16524  Idccid 16525   Func cfunc 16713   ×c cxpc 17007   1stF c1stf 17008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-map 8023  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-dec 11684  df-uz 11878  df-fz 12518  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-hom 16166  df-cco 16167  df-cat 16528  df-cid 16529  df-func 16717  df-xpc 17011  df-1stf 17012
This theorem is referenced by:  prf1st  17043  1st2ndprf  17045  uncfcl  17074  uncf1  17075  uncf2  17076  diagcl  17080  diag11  17082  diag12  17083  diag2  17084  yonedalem1  17111  yonedalem21  17112  yonedalem22  17117
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