Theorem uspgrsprf1 48623

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 48622	. 2 ⊢ 𝐹:𝐺⟶𝑃
5	1, 2, 3	uspgrsprfv 48621	. . . . 5 ⊢ (𝑎 ∈ 𝐺 → (𝐹‘𝑎) = (2nd ‘𝑎))
6	1, 2, 3	uspgrsprfv 48621	. . . . 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 2828	. . . . . 6 ⊢ (𝑎 ∈ 𝐺 ↔ 𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
9		elopab 5482	. . . . . 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 4818	. . . . . . . . 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 480	. . . . . . . . 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 639	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
17	16	rexbidv 3161	. . . . . . . . 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 633	. . . . . . . 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 633	. . . . . . 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 2043	. . . . . 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 297	. . . . 5 ⊢ (𝑎 ∈ 𝐺 ↔ ∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
22	2	eleq2i 2828	. . . . . 6 ⊢ (𝑏 ∈ 𝐺 ↔ 𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
23		elopab 5482	. . . . . 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 275	. . . . 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 4818	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑣 ∧ 𝑓 = 𝑒) → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)
27	26	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑣 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
28	27	equcoms 2022	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
29	25, 28	biimtrrdi 254	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑉 → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
30	29	ad2antrl 729	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 3433	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
35		vex 3433	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
36	34, 35	op2ndd 7953	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (2nd ‘𝑎) = 𝑓)
37	36	ad2antrl 729	. . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
39		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
40	38, 39	op2ndd 7953	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ⟨𝑣, 𝑒⟩ → (2nd ‘𝑏) = 𝑒)
41	40	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑏) = 𝑒)
42	41	adantr 480	. . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (𝑏 = ⟨𝑣, 𝑒⟩ → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
46	45	adantr 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 294	. . . . . . . . . 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 412	. . . . . . . . 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 1934	. . . . . . . 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 1934	. . . . . 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 406	. . . . 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 599	. . . 4 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏))
57	7, 56	sylbid 240	. . 3 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
58	57	rgen2 3177	. 2 ⊢ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
59		dff13 7209	. 2 ⊢ (𝐹:𝐺–1-1→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
60	4, 58, 59	mpbir2an 712	1 ⊢ 𝐹:𝐺–1-1→𝑃

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  𝒫 cpw 4541  ⟨cop 4573  {copab 5147   ↦ cmpt 5166  ⟶wf 6494  –1-1→wf1 6495  ‘cfv 6498  2^nd c2nd 7941  Vtxcvtx 29065  Edgcedg 29116  USPGraphcuspgr 29217  Pairscspr 47937

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-edg 29117  df-upgr 29151  df-uspgr 29219  df-spr 47938

This theorem is referenced by:  uspgrsprf1o  48625