Theorem uspgrsprf1 48089

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 48088	. 2 ⊢ 𝐹:𝐺⟶𝑃
5	1, 2, 3	uspgrsprfv 48087	. . . . 5 ⊢ (𝑎 ∈ 𝐺 → (𝐹‘𝑎) = (2nd ‘𝑎))
6	1, 2, 3	uspgrsprfv 48087	. . . . 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 2827	. . . . . 6 ⊢ (𝑎 ∈ 𝐺 ↔ 𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
9		elopab 5507	. . . . . 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 4856	. . . . . . . . 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 638	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
17	16	rexbidv 3165	. . . . . . . . 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 632	. . . . . . . 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 632	. . . . . . 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 2041	. . . . . 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 2827	. . . . . 6 ⊢ (𝑏 ∈ 𝐺 ↔ 𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
23		elopab 5507	. . . . . 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 4856	. . . . . . . . . . . . . . . . . 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 2020	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 3468	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
35		vex 3468	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
36	34, 35	op2ndd 8004	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (2nd ‘𝑎) = 𝑓)
37	36	ad2antrl 728	. . . . . . . . . . . 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 3468	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
39		vex 3468	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
40	38, 39	op2ndd 8004	. . . . . . . . . . . . . 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 1932	. . . . . . . 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 1932	. . . . . 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 598	. . . 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 3185	. 2 ⊢ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
59		dff13 7252	. 2 ⊢ (𝐹:𝐺–1-1→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
60	4, 58, 59	mpbir2an 711	1 ⊢ 𝐹:𝐺–1-1→𝑃

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  𝒫 cpw 4580  ⟨cop 4612  {copab 5186   ↦ cmpt 5206  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  2^nd c2nd 7992  Vtxcvtx 28980  Edgcedg 29031  USPGraphcuspgr 29132  Pairscspr 47458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9920  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-fz 13530  df-hash 14354  df-edg 29032  df-upgr 29066  df-uspgr 29134  df-spr 47459

This theorem is referenced by:  uspgrsprf1o  48091