Theorem uspgrsprf1 46039

Description: The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)

Hypotheses

Ref	Expression
uspgrsprf.p	⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
uspgrsprf.g	⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
uspgrsprf.f	⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))

Assertion

Ref	Expression
uspgrsprf1	⊢ 𝐹:𝐺–1-1→𝑃

Distinct variable groups:   𝑃,𝑒,𝑞,𝑣   𝑒,𝑉,𝑞,𝑣   𝑔,𝐺   𝑃,𝑔,𝑒,𝑣

Allowed substitution hints:   𝐹(𝑣,𝑒,𝑔,𝑞)   𝐺(𝑣,𝑒,𝑞)   𝑉(𝑔)

Proof of Theorem uspgrsprf1

Dummy variables 𝑎 𝑏 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrsprf.p	. . 3 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
2		uspgrsprf.g	. . 3 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
3		uspgrsprf.f	. . 3 ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
4	1, 2, 3	uspgrsprf 46038	. 2 ⊢ 𝐹:𝐺⟶𝑃
5	1, 2, 3	uspgrsprfv 46037	. . . . 5 ⊢ (𝑎 ∈ 𝐺 → (𝐹‘𝑎) = (2nd ‘𝑎))
6	1, 2, 3	uspgrsprfv 46037	. . . . 5 ⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏))
7	5, 6	eqeqan12d 2750	. . . 4 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (2nd ‘𝑎) = (2nd ‘𝑏)))
8	2	eleq2i 2829	. . . . . 6 ⊢ (𝑎 ∈ 𝐺 ↔ 𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
9		elopab 5484	. . . . . 6 ⊢ (𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣∃𝑒(𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
10		opeq12 4832	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ⟨𝑣, 𝑒⟩ = ⟨𝑤, 𝑓⟩)
11	10	eqeq2d 2747	. . . . . . . 8 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑎 = ⟨𝑣, 𝑒⟩ ↔ 𝑎 = ⟨𝑤, 𝑓⟩))
12		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑣 = 𝑤 → (𝑣 = 𝑉 ↔ 𝑤 = 𝑉))
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣 = 𝑉 ↔ 𝑤 = 𝑉))
14		eqeq2 2748	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑤))
15		eqeq2 2748	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑓))
16	14, 15	bi2anan9 637	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
17	16	rexbidv 3175	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
18	13, 17	anbi12d 631	. . . . . . . 8 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
19	11, 18	anbi12d 631	. . . . . . 7 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ↔ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))))
20	19	cbvex2vw 2044	. . . . . 6 ⊢ (∃𝑣∃𝑒(𝑎 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ↔ ∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
21	8, 9, 20	3bitri 296	. . . . 5 ⊢ (𝑎 ∈ 𝐺 ↔ ∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
22	2	eleq2i 2829	. . . . . 6 ⊢ (𝑏 ∈ 𝐺 ↔ 𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
23		elopab 5484	. . . . . 6 ⊢ (𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
24	22, 23	bitri 274	. . . . 5 ⊢ (𝑏 ∈ 𝐺 ↔ ∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
25		eqeq2 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑉 → (𝑣 = 𝑤 ↔ 𝑣 = 𝑉))
26		opeq12 4832	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑣 ∧ 𝑓 = 𝑒) → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)
27	26	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑣 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
28	27	equcoms 2023	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
29	25, 28	syl6bir 253	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑉 → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
30	29	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
31	30	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑉 → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
32	31	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
33	32	imp 407	. . . . . . . . . . 11 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
34		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
35		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
36	34, 35	op2ndd 7932	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (2nd ‘𝑎) = 𝑓)
37	36	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (2nd ‘𝑎) = 𝑓)
38		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
39		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
40	38, 39	op2ndd 7932	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ⟨𝑣, 𝑒⟩ → (2nd ‘𝑏) = 𝑒)
41	40	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑏) = 𝑒)
42	41	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (2nd ‘𝑏) = 𝑒)
43	37, 42	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → ((2nd ‘𝑎) = (2nd ‘𝑏) ↔ 𝑓 = 𝑒))
44		eqeq12 2753	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ 𝑏 = ⟨𝑣, 𝑒⟩) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
45	44	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (𝑏 = ⟨𝑣, 𝑒⟩ → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
46	45	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑏 = ⟨𝑣, 𝑒⟩ → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
47	46	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑏 = ⟨𝑣, 𝑒⟩ → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
48	47	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
49	48	imp 407	. . . . . . . . . . 11 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
50	33, 43, 49	3imtr4d 293	. . . . . . . . . 10 ⊢ (((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) ∧ (𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏))
51	50	ex 413	. . . . . . . . 9 ⊢ ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏)))
52	51	exlimivv 1935	. . . . . . . 8 ⊢ (∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏)))
53	52	com12 32	. . . . . . 7 ⊢ ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏)))
54	53	exlimivv 1935	. . . . . 6 ⊢ (∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) → (∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏)))
55	54	imp 407	. . . . 5 ⊢ ((∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))) ∧ ∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)))) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏))
56	21, 24, 55	syl2anb 598	. . . 4 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏))
57	7, 56	sylbid 239	. . 3 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
58	57	rgen2 3194	. 2 ⊢ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
59		dff13 7202	. 2 ⊢ (𝐹:𝐺–1-1→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
60	4, 58, 59	mpbir2an 709	1 ⊢ 𝐹:𝐺–1-1→𝑃

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  𝒫 cpw 4560  ⟨cop 4592  {copab 5167   ↦ cmpt 5188  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496  2^nd c2nd 7920  Vtxcvtx 27947  Edgcedg 27998  USPGraphcuspgr 28099  Pairscspr 45659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-upgr 28033  df-uspgr 28101  df-spr 45660

This theorem is referenced by:  uspgrsprf1o  46041