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Theorem xpcco 17030
 Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpccofval.t 𝑇 = (𝐶 ×c 𝐷)
xpccofval.b 𝐵 = (Base‘𝑇)
xpccofval.k 𝐾 = (Hom ‘𝑇)
xpccofval.o1 · = (comp‘𝐶)
xpccofval.o2 = (comp‘𝐷)
xpccofval.o 𝑂 = (comp‘𝑇)
xpcco.x (𝜑𝑋𝐵)
xpcco.y (𝜑𝑌𝐵)
xpcco.z (𝜑𝑍𝐵)
xpcco.f (𝜑𝐹 ∈ (𝑋𝐾𝑌))
xpcco.g (𝜑𝐺 ∈ (𝑌𝐾𝑍))
Assertion
Ref Expression
xpcco (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩)

Proof of Theorem xpcco
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccofval.t . . 3 𝑇 = (𝐶 ×c 𝐷)
2 xpccofval.b . . 3 𝐵 = (Base‘𝑇)
3 xpccofval.k . . 3 𝐾 = (Hom ‘𝑇)
4 xpccofval.o1 . . 3 · = (comp‘𝐶)
5 xpccofval.o2 . . 3 = (comp‘𝐷)
6 xpccofval.o . . 3 𝑂 = (comp‘𝑇)
71, 2, 3, 4, 5, 6xpccofval 17029 . 2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))
8 xpcco.x . . . 4 (𝜑𝑋𝐵)
9 xpcco.y . . . 4 (𝜑𝑌𝐵)
10 opelxpi 5288 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 565 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 xpcco.z . . . 4 (𝜑𝑍𝐵)
1312adantr 466 . . 3 ((𝜑𝑥 = ⟨𝑋, 𝑌⟩) → 𝑍𝐵)
14 ovex 6822 . . . . 5 ((2nd𝑥)𝐾𝑦) ∈ V
15 fvex 6342 . . . . 5 (𝐾𝑥) ∈ V
1614, 15mpt2ex 7396 . . . 4 (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩) ∈ V
1716a1i 11 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩) ∈ V)
18 xpcco.g . . . . . 6 (𝜑𝐺 ∈ (𝑌𝐾𝑍))
1918adantr 466 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ (𝑌𝐾𝑍))
20 simprl 746 . . . . . . . 8 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2120fveq2d 6336 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
22 op2ndg 7327 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
238, 9, 22syl2anc 565 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2423adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2521, 24eqtrd 2804 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = 𝑌)
26 simprr 748 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
2725, 26oveq12d 6810 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2nd𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
2819, 27eleqtrrd 2852 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ ((2nd𝑥)𝐾𝑦))
29 xpcco.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐾𝑌))
3029adantr 466 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝑋𝐾𝑌))
3120fveq2d 6336 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
32 df-ov 6795 . . . . . . 7 (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
3331, 32syl6eqr 2822 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝑋𝐾𝑌))
3430, 33eleqtrrd 2852 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝐾𝑥))
3534adantr 466 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝐾𝑥))
36 opex 5060 . . . . 5 ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩ ∈ V
3736a1i 11 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩ ∈ V)
3820fveq2d 6336 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
39 op1stg 7326 . . . . . . . . . . . . 13 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
408, 9, 39syl2anc 565 . . . . . . . . . . . 12 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4140adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4238, 41eqtrd 2804 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = 𝑋)
4342adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑥) = 𝑋)
4443fveq2d 6336 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(1st𝑥)) = (1st𝑋))
4525adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑥) = 𝑌)
4645fveq2d 6336 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(2nd𝑥)) = (1st𝑌))
4744, 46opeq12d 4545 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ = ⟨(1st𝑋), (1st𝑌)⟩)
48 simplrr 755 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑦 = 𝑍)
4948fveq2d 6336 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑦) = (1st𝑍))
5047, 49oveq12d 6810 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦)) = (⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍)))
51 simprl 746 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
5251fveq2d 6336 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
53 simprr 748 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
5453fveq2d 6336 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
5550, 52, 54oveq123d 6813 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
5643fveq2d 6336 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(1st𝑥)) = (2nd𝑋))
5745fveq2d 6336 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(2nd𝑥)) = (2nd𝑌))
5856, 57opeq12d 4545 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ = ⟨(2nd𝑋), (2nd𝑌)⟩)
5948fveq2d 6336 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑦) = (2nd𝑍))
6058, 59oveq12d 6810 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦)) = (⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍)))
6151fveq2d 6336 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
6253fveq2d 6336 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
6360, 61, 62oveq123d 6813 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹)))
6455, 63opeq12d 4545 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩ = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩)
6528, 35, 37, 64ovmpt2dv2 6940 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩) → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩))
6611, 13, 17, 65ovmpt2dv 6939 . 2 (𝜑 → (𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩))
677, 66mpi 20 1 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ⟨cop 4320   × cxp 5247  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  1st c1st 7312  2nd c2nd 7313  Basecbs 16063  Hom chom 16159  compcco 16160   ×c cxpc 17015 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 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-hom 16173  df-cco 16174  df-xpc 17019 This theorem is referenced by:  xpcco1st  17031  xpcco2nd  17032  xpcco2  17034  xpccatid  17035
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