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Theorem xpcco 17209
 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 17208 . 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)
108, 9opelxpd 5393 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 xpcco.z . . . 4 (𝜑𝑍𝐵)
1211adantr 474 . . 3 ((𝜑𝑥 = ⟨𝑋, 𝑌⟩) → 𝑍𝐵)
13 ovex 6954 . . . . 5 ((2nd𝑥)𝐾𝑦) ∈ V
14 fvex 6459 . . . . 5 (𝐾𝑥) ∈ V
1513, 14mpt2ex 7527 . . . 4 (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩) ∈ V
1615a1i 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)
17 xpcco.g . . . . . 6 (𝜑𝐺 ∈ (𝑌𝐾𝑍))
1817adantr 474 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ (𝑌𝐾𝑍))
19 simprl 761 . . . . . . . 8 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
2019fveq2d 6450 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
21 op2ndg 7458 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
228, 9, 21syl2anc 579 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2322adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2420, 23eqtrd 2814 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (2nd𝑥) = 𝑌)
25 simprr 763 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝑦 = 𝑍)
2624, 25oveq12d 6940 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → ((2nd𝑥)𝐾𝑦) = (𝑌𝐾𝑍))
2718, 26eleqtrrd 2862 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐺 ∈ ((2nd𝑥)𝐾𝑦))
28 xpcco.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐾𝑌))
2928adantr 474 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝑋𝐾𝑌))
3019fveq2d 6450 . . . . . . 7 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝐾‘⟨𝑋, 𝑌⟩))
31 df-ov 6925 . . . . . . 7 (𝑋𝐾𝑌) = (𝐾‘⟨𝑋, 𝑌⟩)
3230, 31syl6eqr 2832 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (𝐾𝑥) = (𝑋𝐾𝑌))
3329, 32eleqtrrd 2862 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → 𝐹 ∈ (𝐾𝑥))
3433adantr 474 . . . 4 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝐾𝑥))
35 opex 5164 . . . . 5 ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩ ∈ V
3635a1i 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)
3719fveq2d 6450 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
38 op1stg 7457 . . . . . . . . . . . . 13 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
398, 9, 38syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4039adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4137, 40eqtrd 2814 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) → (1st𝑥) = 𝑋)
4241adantr 474 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑥) = 𝑋)
4342fveq2d 6450 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(1st𝑥)) = (1st𝑋))
4424adantr 474 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑥) = 𝑌)
4544fveq2d 6450 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st ‘(2nd𝑥)) = (1st𝑌))
4643, 45opeq12d 4644 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ = ⟨(1st𝑋), (1st𝑌)⟩)
47 simplrr 768 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑦 = 𝑍)
4847fveq2d 6450 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑦) = (1st𝑍))
4946, 48oveq12d 6940 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦)) = (⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍)))
50 simprl 761 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
5150fveq2d 6450 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑔) = (1st𝐺))
52 simprr 763 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
5352fveq2d 6450 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (1st𝑓) = (1st𝐹))
5449, 51, 53oveq123d 6943 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
5542fveq2d 6450 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(1st𝑥)) = (2nd𝑋))
5644fveq2d 6450 . . . . . . . 8 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd ‘(2nd𝑥)) = (2nd𝑌))
5755, 56opeq12d 4644 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ = ⟨(2nd𝑋), (2nd𝑌)⟩)
5847fveq2d 6450 . . . . . . 7 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑦) = (2nd𝑍))
5957, 58oveq12d 6940 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦)) = (⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍)))
6050fveq2d 6450 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
6152fveq2d 6450 . . . . . 6 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
6259, 60, 61oveq123d 6943 . . . . 5 (((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = 𝑍)) ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹)))
6354, 62opeq12d 4644 . . . 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))⟩)
6427, 34, 36, 63ovmpt2dv2 7071 . . 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))⟩))
6510, 12, 16, 64ovmpt2dv 7070 . 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))⟩))
667, 65mpi 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 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404   × cxp 5353  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1st c1st 7443  2nd c2nd 7444  Basecbs 16255  Hom chom 16349  compcco 16350   ×c cxpc 17194 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-hom 16362  df-cco 16363  df-xpc 17198 This theorem is referenced by:  xpcco1st  17210  xpcco2nd  17211  xpcco2  17213  xpccatid  17214
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