Theorem uspgrsprf1 41755

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 41754	. 2 ⊢ 𝐹:𝐺⟶𝑃
5	1, 2, 3	uspgrsprfv 41753	. . . . 5 ⊢ (𝑎 ∈ 𝐺 → (𝐹‘𝑎) = (2nd ‘𝑎))
6	1, 2, 3	uspgrsprfv 41753	. . . . 5 ⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏))
7	5, 6	eqeqan12d 2638	. . . 4 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (2nd ‘𝑎) = (2nd ‘𝑏)))
8	2	eleq2i 2693	. . . . . 6 ⊢ (𝑎 ∈ 𝐺 ↔ 𝑎 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
9		elopab 4983	. . . . . 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 4404	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ⟨𝑣, 𝑒⟩ = ⟨𝑤, 𝑓⟩)
11	10	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑎 = ⟨𝑣, 𝑒⟩ ↔ 𝑎 = ⟨𝑤, 𝑓⟩))
12		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑣 = 𝑤 → (𝑣 = 𝑉 ↔ 𝑤 = 𝑉))
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣 = 𝑉 ↔ 𝑤 = 𝑉))
14		eqeq2 2633	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑤))
15		eqeq2 2633	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑓 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑓))
16	14, 15	bi2anan9 917	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓)))
17	16	rexbidv 3052	. . . . . . . . 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 747	. . . . . . . 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 747	. . . . . . 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	cbvex2v 2287	. . . . . 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 286	. . . . 5 ⊢ (𝑎 ∈ 𝐺 ↔ ∃𝑤∃𝑓(𝑎 = ⟨𝑤, 𝑓⟩ ∧ (𝑤 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑤 ∧ (Edg‘𝑞) = 𝑓))))
22	2	eleq2i 2693	. . . . . 6 ⊢ (𝑏 ∈ 𝐺 ↔ 𝑏 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
23		elopab 4983	. . . . . 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 264	. . . . 5 ⊢ (𝑏 ∈ 𝐺 ↔ ∃𝑣∃𝑒(𝑏 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
25		eqeq2 2633	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑉 → (𝑣 = 𝑤 ↔ 𝑣 = 𝑉))
26		opeq12 4404	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑣 ∧ 𝑓 = 𝑒) → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)
27	26	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑣 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
28	27	equcoms 1947	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
29	25, 28	syl6bir 244	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑉 → (𝑣 = 𝑉 → (𝑓 = 𝑒 → ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩)))
30	29	ad2antrl 764	. . . . . . . . . . . . . 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 764	. . . . . . . . . . . 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 445	. . . . . . . . . . 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 3203	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
35		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
36	34, 35	op2ndd 7179	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑤, 𝑓⟩ → (2nd ‘𝑎) = 𝑓)
37	36	ad2antrl 764	. . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
39		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
40	38, 39	op2ndd 7179	. . . . . . . . . . . . . 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 2637	. . . . . . . . . . 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 2635	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = ⟨𝑤, 𝑓⟩ ∧ 𝑏 = ⟨𝑣, 𝑒⟩) → (𝑎 = 𝑏 ↔ ⟨𝑤, 𝑓⟩ = ⟨𝑣, 𝑒⟩))
45	44	ex 450	. . . . . . . . . . . . . . 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 445	. . . . . . . . . . 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 283	. . . . . . . . . 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 450	. . . . . . . . 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 1860	. . . . . . . 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 1860	. . . . . 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 445	. . . . 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 496	. . . 4 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((2nd ‘𝑎) = (2nd ‘𝑏) → 𝑎 = 𝑏))
57	7, 56	sylbid 230	. . 3 ⊢ ((𝑎 ∈ 𝐺 ∧ 𝑏 ∈ 𝐺) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
58	57	rgen2a 2977	. 2 ⊢ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
59		dff13 6512	. 2 ⊢ (𝐹:𝐺–1-1→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝐺 ∀𝑏 ∈ 𝐺 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
60	4, 58, 59	mpbir2an 955	1 ⊢ 𝐹:𝐺–1-1→𝑃

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  𝒫 cpw 4158  ⟨cop 4183  {copab 4712   ↦ cmpt 4729  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  2^nd c2nd 7167  Vtxcvtx 25874  Edgcedg 25939   USPGraph cuspgr 26043  Pairscspr 41727

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-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-rmo 2920  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-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977  df-uspgr 26045  df-spr 41728

This theorem is referenced by:  uspgrsprf1o  41757