Theorem fcnvgreu 32691

Description: If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.)

Assertion

Ref	Expression
fcnvgreu	⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝))

Distinct variable groups:   𝐴,𝑝   𝑌,𝑝

Proof of Theorem fcnvgreu

Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rn 5711	. . . 4 ⊢ ran 𝐴 = dom ◡𝐴
2	1	eleq2i 2836	. . 3 ⊢ (𝑌 ∈ ran 𝐴 ↔ 𝑌 ∈ dom ◡𝐴)
3		fgreu 32690	. . . 4 ⊢ ((Fun ◡𝐴 ∧ 𝑌 ∈ dom ◡𝐴) → ∃!𝑞 ∈ ◡ 𝐴𝑌 = (1st ‘𝑞))
4	3	adantll 713	. . 3 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ dom ◡𝐴) → ∃!𝑞 ∈ ◡ 𝐴𝑌 = (1st ‘𝑞))
5	2, 4	sylan2b 593	. 2 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 ∈ ◡ 𝐴𝑌 = (1st ‘𝑞))
6		cnvcnvss 6225	. . . . . 6 ⊢ ◡◡𝐴 ⊆ 𝐴
7		cnvssrndm 6302	. . . . . . . . . . 11 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)
8	7	sseli 4004	. . . . . . . . . 10 ⊢ (𝑞 ∈ ◡𝐴 → 𝑞 ∈ (ran 𝐴 × dom 𝐴))
9		dfdm4 5920	. . . . . . . . . . 11 ⊢ dom 𝐴 = ran ◡𝐴
10	1, 9	xpeq12i 5728	. . . . . . . . . 10 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
11	8, 10	eleqtrdi 2854	. . . . . . . . 9 ⊢ (𝑞 ∈ ◡𝐴 → 𝑞 ∈ (dom ◡𝐴 × ran ◡𝐴))
12		2nd1st 8079	. . . . . . . . 9 ⊢ (𝑞 ∈ (dom ◡𝐴 × ran ◡𝐴) → ∪ ◡{𝑞} = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝑞 ∈ ◡𝐴 → ∪ ◡{𝑞} = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
14	13	eqcomd 2746	. . . . . . 7 ⊢ (𝑞 ∈ ◡𝐴 → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞})
15		relcnv 6134	. . . . . . . 8 ⊢ Rel ◡𝐴
16		cnvf1olem 8151	. . . . . . . . 9 ⊢ ((Rel ◡𝐴 ∧ (𝑞 ∈ ◡𝐴 ∧ ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞})) → (⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ ◡◡𝐴 ∧ 𝑞 = ∪ ◡{⟨(2nd ‘𝑞), (1st ‘𝑞)⟩}))
17	16	simpld 494	. . . . . . . 8 ⊢ ((Rel ◡𝐴 ∧ (𝑞 ∈ ◡𝐴 ∧ ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞})) → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ ◡◡𝐴)
18	15, 17	mpan 689	. . . . . . 7 ⊢ ((𝑞 ∈ ◡𝐴 ∧ ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞}) → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ ◡◡𝐴)
19	14, 18	mpdan 686	. . . . . 6 ⊢ (𝑞 ∈ ◡𝐴 → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ ◡◡𝐴)
20	6, 19	sselid 4006	. . . . 5 ⊢ (𝑞 ∈ ◡𝐴 → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ 𝐴)
21	20	adantl 481	. . . 4 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑞 ∈ ◡𝐴) → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ∈ 𝐴)
22		simpll 766	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → Rel 𝐴)
23		simpr 484	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴)
24		relssdmrn 6299	. . . . . . . . . . 11 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((Rel 𝐴 ∧ Fun ◡𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
26	25	sselda 4008	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27		2nd1st 8079	. . . . . . . . 9 ⊢ (𝑝 ∈ (dom 𝐴 × ran 𝐴) → ∪ ◡{𝑝} = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)
28	26, 27	syl 17	. . . . . . . 8 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ∪ ◡{𝑝} = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)
29	28	eqcomd 2746	. . . . . . 7 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ = ∪ ◡{𝑝})
30		cnvf1olem 8151	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ (𝑝 ∈ 𝐴 ∧ ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ = ∪ ◡{𝑝})) → (⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ ∈ ◡𝐴 ∧ 𝑝 = ∪ ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩}))
31	30	simpld 494	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝑝 ∈ 𝐴 ∧ ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ = ∪ ◡{𝑝})) → ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ ∈ ◡𝐴)
32	22, 23, 29, 31	syl12anc 836	. . . . . 6 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ ∈ ◡𝐴)
33	15	a1i 11	. . . . . . . . . 10 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → Rel ◡𝐴)
34		simplr 768	. . . . . . . . . 10 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → 𝑞 ∈ ◡𝐴)
35	14	ad2antlr 726	. . . . . . . . . 10 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞})
36	16	simprd 495	. . . . . . . . . 10 ⊢ ((Rel ◡𝐴 ∧ (𝑞 ∈ ◡𝐴 ∧ ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ = ∪ ◡{𝑞})) → 𝑞 = ∪ ◡{⟨(2nd ‘𝑞), (1st ‘𝑞)⟩})
37	33, 34, 35, 36	syl12anc 836	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → 𝑞 = ∪ ◡{⟨(2nd ‘𝑞), (1st ‘𝑞)⟩})
38		simpr 484	. . . . . . . . . . . 12 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
39	38	sneqd 4660	. . . . . . . . . . 11 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → {𝑝} = {⟨(2nd ‘𝑞), (1st ‘𝑞)⟩})
40	39	cnveqd 5900	. . . . . . . . . 10 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → ◡{𝑝} = ◡{⟨(2nd ‘𝑞), (1st ‘𝑞)⟩})
41	40	unieqd 4944	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → ∪ ◡{𝑝} = ∪ ◡{⟨(2nd ‘𝑞), (1st ‘𝑞)⟩})
42	28	ad2antrr 725	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → ∪ ◡{𝑝} = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)
43	37, 41, 42	3eqtr2d 2786	. . . . . . . 8 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)
44	30	simprd 495	. . . . . . . . . . 11 ⊢ ((Rel 𝐴 ∧ (𝑝 ∈ 𝐴 ∧ ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ = ∪ ◡{𝑝})) → 𝑝 = ∪ ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
45	22, 23, 29, 44	syl12anc 836	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → 𝑝 = ∪ ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
46	45	ad2antrr 725	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → 𝑝 = ∪ ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
47		simpr 484	. . . . . . . . . . . 12 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)
48	47	sneqd 4660	. . . . . . . . . . 11 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → {𝑞} = {⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
49	48	cnveqd 5900	. . . . . . . . . 10 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → ◡{𝑞} = ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
50	49	unieqd 4944	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → ∪ ◡{𝑞} = ∪ ◡{⟨(2nd ‘𝑝), (1st ‘𝑝)⟩})
51	13	ad2antlr 726	. . . . . . . . 9 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → ∪ ◡{𝑞} = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
52	46, 50, 51	3eqtr2d 2786	. . . . . . . 8 ⊢ (((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) ∧ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩) → 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
53	43, 52	impbida 800	. . . . . . 7 ⊢ ((((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ ◡𝐴) → (𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩))
54	53	ralrimiva 3152	. . . . . 6 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩))
55		eqeq2 2752	. . . . . . . . 9 ⊢ (𝑟 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ → (𝑞 = 𝑟 ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩))
56	55	bibi2d 342	. . . . . . . 8 ⊢ (𝑟 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ → ((𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)))
57	56	ralbidv 3184	. . . . . . 7 ⊢ (𝑟 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ → (∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)))
58	57	rspcev 3635	. . . . . 6 ⊢ ((⟨(2nd ‘𝑝), (1st ‘𝑝)⟩ ∈ ◡𝐴 ∧ ∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = ⟨(2nd ‘𝑝), (1st ‘𝑝)⟩)) → ∃𝑟 ∈ ◡ 𝐴∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = 𝑟))
59	32, 54, 58	syl2anc 583	. . . . 5 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ∃𝑟 ∈ ◡ 𝐴∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = 𝑟))
60		reu6 3748	. . . . 5 ⊢ (∃!𝑞 ∈ ◡ 𝐴𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ ∃𝑟 ∈ ◡ 𝐴∀𝑞 ∈ ◡ 𝐴(𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ ↔ 𝑞 = 𝑟))
61	59, 60	sylibr 234	. . . 4 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 ∈ 𝐴) → ∃!𝑞 ∈ ◡ 𝐴𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩)
62		fvex 6933	. . . . . . 7 ⊢ (2nd ‘𝑞) ∈ V
63		fvex 6933	. . . . . . 7 ⊢ (1st ‘𝑞) ∈ V
64	62, 63	op2ndd 8041	. . . . . 6 ⊢ (𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ → (2nd ‘𝑝) = (1st ‘𝑞))
65	64	eqeq2d 2751	. . . . 5 ⊢ (𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩ → (𝑌 = (2nd ‘𝑝) ↔ 𝑌 = (1st ‘𝑞)))
66	65	adantl 481	. . . 4 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑝 = ⟨(2nd ‘𝑞), (1st ‘𝑞)⟩) → (𝑌 = (2nd ‘𝑝) ↔ 𝑌 = (1st ‘𝑞)))
67	21, 61, 66	reuxfr1d 3772	. . 3 ⊢ ((Rel 𝐴 ∧ Fun ◡𝐴) → (∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝) ↔ ∃!𝑞 ∈ ◡ 𝐴𝑌 = (1st ‘𝑞)))
68	67	adantr 480	. 2 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ ran 𝐴) → (∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝) ↔ ∃!𝑞 ∈ ◡ 𝐴𝑌 = (1st ‘𝑞)))
69	5, 68	mpbird 257	1 ⊢ (((Rel 𝐴 ∧ Fun ◡𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ cuni 4931   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701  Rel wrel 5705  Fun wfun 6567  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-1st 8030  df-2nd 8031

This theorem is referenced by:  gsummpt2co  33031