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Theorem xpccofval 16822

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)

**Assertion**
Ref	Expression
xpccofval	⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))

Distinct variable groups: 𝑓,𝑔,𝑥,𝑦,𝐵 𝐶,𝑓,𝑔,𝑥,𝑦 𝐷,𝑓,𝑔,𝑥,𝑦 · ,𝑓,𝑔,𝑥,𝑦 ∙ ,𝑓,𝑔,𝑥,𝑦 𝑓,𝐾,𝑔,𝑥,𝑦 𝑥,𝑂,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑓,𝑔) 𝑂(𝑓,𝑔)

Proof of Theorem xpccofval Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccofval.t	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		eqid 2622	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2622	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2622	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2622	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		xpccofval.o1	. . . 4 ⊢ · = (comp‘𝐶)
7		xpccofval.o2	. . . 4 ⊢ ∙ = (comp‘𝐷)
8		simpl 473	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 477	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		xpccofval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
11	1, 2, 3	xpcbas 16818	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
12	10, 11	eqtr4i 2647	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
13	12	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14		xpccofval.k	. . . . . 6 ⊢ 𝐾 = (Hom ‘𝑇)
15	1, 10, 4, 5, 14	xpchomfval 16819	. . . . 5 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
16	15	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))))
17		eqidd 2623	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17	xpcval 16817	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
19		catstr 16617	. . 3 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} Struct ⟨1, ;15⟩
20		ccoid 16077	. . 3 ⊢ comp = Slot (comp‘ndx)
21		snsstp3 4349	. . 3 ⊢ {⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
22		fvex 6201	. . . . . . 7 ⊢ (Base‘𝑇) ∈ V
23	10, 22	eqeltri 2697	. . . . . 6 ⊢ 𝐵 ∈ V
24	23, 23	xpex 6962	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
25	24, 23	mpt2ex 7247	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) ∈ V
26	25	a1i 11	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) ∈ V)
27		xpccofval.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
28	18, 19, 20, 21, 26, 27	strfv3 15908	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
29		mpt20 6725	. . . 4 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = ∅
30	29	eqcomi 2631	. . 3 ⊢ ∅ = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
31		fnxpc 16816	. . . . . . . 8 ⊢ ×_c Fn (V × V)
32		fndm 5990	. . . . . . . 8 ⊢ ( ×_c Fn (V × V) → dom ×_c = (V × V))
33	31, 32	ax-mp 5	. . . . . . 7 ⊢ dom ×_c = (V × V)
34	33	ndmov 6818	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
35	1, 34	syl5eq 2668	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
36	35	fveq2d 6195	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (comp‘𝑇) = (comp‘∅))
37	20	str0 15911	. . . 4 ⊢ ∅ = (comp‘∅)
38	36, 27, 37	3eqtr4g 2681	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = ∅)
39	35	fveq2d 6195	. . . . . . 7 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
40		base0 15912	. . . . . . 7 ⊢ ∅ = (Base‘∅)
41	39, 10, 40	3eqtr4g 2681	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
42	41	xpeq2d 5139	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
43		xp0 5552	. . . . 5 ⊢ (𝐵 × ∅) = ∅
44	42, 43	syl6eq 2672	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 × 𝐵) = ∅)
45		eqidd 2623	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩) = (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
46	44, 41, 45	mpt2eq123dv 6717	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
47	30, 38, 46	3eqtr4a 2682	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
48	28, 47	pm2.61i 176	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {ctp 4181 ⟨cop 4183 × cxp 5112 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 1c1 9937 5c5 11073 cdc 11493 ndxcnx 15854 Basecbs 15857 Hom chom 15952 compcco 15953 ×_c cxpc 16808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-xpc 16812

This theorem is referenced by: xpcco 16823

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