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Theorem xpcco 17044
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 17043 . 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 5305 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
118, 9, 10syl2anc 696 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
12 xpcco.z . . . 4 (𝜑𝑍𝐵)
1312adantr 472 . . 3 ((𝜑𝑥 = ⟨𝑋, 𝑌⟩) → 𝑍𝐵)
14 ovex 6842 . . . . 5 ((2nd𝑥)𝐾𝑦) ∈ V
15 fvex 6363 . . . . 5 (𝐾𝑥) ∈ V
1614, 15mpt2ex 7416 . . . 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 472 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ (𝑌𝐾𝑍))
20 simprl 811 . . . . . . . 8 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2120fveq2d 6357 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
22 op2ndg 7347 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
238, 9, 22syl2anc 696 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2423adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2521, 24eqtrd 2794 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = 𝑌)
26 simprr 813 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
2725, 26oveq12d 6832 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2nd𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
2819, 27eleqtrrd 2842 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ ((2nd𝑥)𝐾𝑦))
29 xpcco.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐾𝑌))
3029adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝑋𝐾𝑌))
3120fveq2d 6357 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
32 df-ov 6817 . . . . . . 7 (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
3331, 32syl6eqr 2812 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝑋𝐾𝑌))
3430, 33eleqtrrd 2842 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝐾𝑥))
3534adantr 472 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝐾𝑥))
36 opex 5081 . . . . 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 6357 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
39 op1stg 7346 . . . . . . . . . . . . 13 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
408, 9, 39syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4140adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4238, 41eqtrd 2794 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = 𝑋)
4342adantr 472 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑥) = 𝑋)
4443fveq2d 6357 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(1st𝑥)) = (1st𝑋))
4525adantr 472 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑥) = 𝑌)
4645fveq2d 6357 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(2nd𝑥)) = (1st𝑌))
4744, 46opeq12d 4561 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ = ⟨(1st𝑋), (1st𝑌)⟩)
48 simplrr 820 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑦 = 𝑍)
4948fveq2d 6357 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑦) = (1st𝑍))
5047, 49oveq12d 6832 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦)) = (⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍)))
51 simprl 811 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
5251fveq2d 6357 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
53 simprr 813 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
5453fveq2d 6357 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
5550, 52, 54oveq123d 6835 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
5643fveq2d 6357 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(1st𝑥)) = (2nd𝑋))
5745fveq2d 6357 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(2nd𝑥)) = (2nd𝑌))
5856, 57opeq12d 4561 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ = ⟨(2nd𝑋), (2nd𝑌)⟩)
5948fveq2d 6357 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑦) = (2nd𝑍))
6058, 59oveq12d 6832 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦)) = (⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍)))
6151fveq2d 6357 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
6253fveq2d 6357 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
6360, 61, 62oveq123d 6835 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹)))
6455, 63opeq12d 4561 . . . 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 6960 . . 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 6959 . 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 383   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327   × cxp 5264  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332  2nd c2nd 7333  Basecbs 16079  Hom chom 16174  compcco 16175   ×c cxpc 17029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-xpc 17033
This theorem is referenced by:  xpcco1st  17045  xpcco2nd  17046  xpcco2  17048  xpccatid  17049
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