Theorem uspgrsprfo 48768

Description: The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto 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
uspgrsprfo	⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃)

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

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

Proof of Theorem uspgrsprfo

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

Step	Hyp	Ref	Expression
1		uspgrsprf.p	. . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
2		uspgrsprf.g	. . . 4 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
3		uspgrsprf.f	. . . 4 ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
4	1, 2, 3	uspgrsprf 48766	. . 3 ⊢ 𝐹:𝐺⟶𝑃
5	4	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶𝑃)
6	1	eleq2i 2857	. . . . . . 7 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉))
7		velpw 4563	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
8	6, 7	bitri 278	. . . . . 6 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉))
9		eqidd 2766	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑉 = 𝑉)
10		vex 3461	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ∈ V)
12		f1oi 6849	. . . . . . . . . . . . . . . . 17 ⊢ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):𝑎–1-1-onto→𝑎)
14		dmresi 6045	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ 𝑎) = 𝑎
15		f1oeq2 6799	. . . . . . . . . . . . . . . . 17 ⊢ (dom ( I ↾ 𝑎) = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ↔ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎))
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ↔ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎)
17	13, 16	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎)
18		sprvalpwle2 48093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
19	18	sseq2d 3971	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
20	19	biimpac 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
21	17, 20	jca 520	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22		f1oeq3 6800	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎))
23		sseq1 3964	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
24	22, 23	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑎 → ((( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) ↔ (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})))
25	11, 21, 24	spcedv 3560	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
26		resiexg 7897	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
27	10, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( I ↾ 𝑎) ∈ V
28	27	f11o 7932	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
29	25, 28	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
30	10	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V)
31	30	resiexd 7204	. . . . . . . . . . . . . 14 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
32	31	anim1ci 627	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V))
33		isuspgrop 29420	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
34	32, 33	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
35	29, 34	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
36		fveqeq2 6880	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
37		fveqeq2 6880	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
38	36, 37	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
39	38	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩) → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
40		opvtxfv 29263	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
41	31, 40	sylan2 604	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
42		edgopval 29310	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
43	31, 42	sylan2 604	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
44		rnresi 6068	. . . . . . . . . . . . . 14 ⊢ ran ( I ↾ 𝑎) = 𝑎
45	43, 44	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
46	41, 45	jca 520	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
47	46	ancoms 463	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
48	35, 39, 47	rspcedvd 3586	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
49	9, 48	jca 520	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
50	2	eleq2i 2857	. . . . . . . . . 10 ⊢ (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
51	30	anim1ci 627	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V))
52		eqeq1 2769	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑉 → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
53	52	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
54		eqeq2 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
55		eqeq2 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
56	54, 55	bi2anan9 649	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
57	56	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
58	53, 57	anbi12d 643	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
59	58	opelopabga 5508	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
60	51, 59	syl 18	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
61	50, 60	bitrid 286	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
62	49, 61	mpbird 260	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
63		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
64	63	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
65	64	adantl 486	. . . . . . . 8 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
66		op2ndg 7987	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
67	66	elvd 3463	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
68	67	adantl 486	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
69	68	eqcomd 2771	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
70	62, 65, 69	rspcedvd 3586	. . . . . . 7 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
71	70	ex 417	. . . . . 6 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
72	8, 71	sylbi 220	. . . . 5 ⊢ (𝑎 ∈ 𝑃 → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
73	72	impcom 412	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
74	1, 2, 3	uspgrsprfv 48765	. . . . . . 7 ⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏))
75	74	adantl 486	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝐹‘𝑏) = (2nd ‘𝑏))
76	75	eqeq2d 2776	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝑎 = (𝐹‘𝑏) ↔ 𝑎 = (2nd ‘𝑏)))
77	76	rexbidva 3187	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏) ↔ ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
78	73, 77	mpbird 260	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
79	78	ralrimiva 3157	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
80		dffo3 7087	. 2 ⊢ (𝐹:𝐺–onto→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)))
81	5, 79, 80	sylanbrc 594	1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {csn 4585  ⟨cop 4591   class class class wbr 5105  {copab 5167   ↦ cmpt 5186   I cid 5546  dom cdm 5652  ran crn 5653   ↾ cres 5654  ⟶wf 6521  –1-1→wf1 6522  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525  2^nd c2nd 7973   ≤ cle 11232  2c2 12286  ♯chash 14357  Vtxcvtx 29255  Edgcedg 29306  USPGraphcuspgr 29407  Pairscspr 48081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-n0 12496  df-xnn0 12569  df-z 12583  df-uz 12854  df-fz 13527  df-hash 14358  df-vtx 29257  df-iedg 29258  df-edg 29307  df-upgr 29341  df-uspgr 29409  df-spr 48082

This theorem is referenced by:  uspgrsprf1o  48769