Theorem uspgrsprfo 46605

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 46603	. . 3 ⊢ 𝐹:𝐺⟶𝑃
5	4	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶𝑃)
6	1	eleq2i 2825	. . . . . . 7 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉))
7		velpw 4607	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
8	6, 7	bitri 274	. . . . . 6 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉))
9		eqidd 2733	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑉 = 𝑉)
10		vex 3478	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ∈ V)
12		f1oi 6871	. . . . . . . . . . . . . . . . 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 6051	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ 𝑎) = 𝑎
15		f1oeq2 6822	. . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎)
18		sprvalpwle2 46236	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
19	18	sseq2d 4014	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
20	19	biimpac 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
21	17, 20	jca 512	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
22		f1oeq3 6823	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎))
23		sseq1 4007	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
24	22, 23	anbi12d 631	. . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
26		resiexg 7907	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
27	10, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( I ↾ 𝑎) ∈ V
28	27	f11o 7935	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 7220	. . . . . . . . . . . . . 14 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
32	31	anim1ci 616	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V))
33		isuspgrop 28459	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph)
36		fveqeq2 6900	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
37		fveqeq2 6900	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
38	36, 37	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
39	38	adantl 482	. . . . . . . . . . 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 28302	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
41	31, 40	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
42		edgopval 28349	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
43	31, 42	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
44		rnresi 6074	. . . . . . . . . . . . . 14 ⊢ ran ( I ↾ 𝑎) = 𝑎
45	43, 44	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
46	41, 45	jca 512	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
47	46	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
48	35, 39, 47	rspcedvd 3614	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
49	9, 48	jca 512	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
50	2	eleq2i 2825	. . . . . . . . . 10 ⊢ (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
51	30	anim1ci 616	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V))
52		eqeq1 2736	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑉 → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
53	52	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
54		eqeq2 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
55		eqeq2 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
56	54, 55	bi2anan9 637	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
57	56	rexbidv 3178	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 5533	. . . . . . . . . . 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 17	. . . . . . . . . 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 282	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
62	49, 61	mpbird 256	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
63		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
64	63	eqeq2d 2743	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
65	64	adantl 482	. . . . . . . 8 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
66		op2ndg 7990	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
67	66	elvd 3481	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
68	67	adantl 482	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
69	68	eqcomd 2738	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
70	62, 65, 69	rspcedvd 3614	. . . . . . 7 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
71	70	ex 413	. . . . . 6 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
72	8, 71	sylbi 216	. . . . 5 ⊢ (𝑎 ∈ 𝑃 → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
73	72	impcom 408	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
74	1, 2, 3	uspgrsprfv 46602	. . . . . . 7 ⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏))
75	74	adantl 482	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝐹‘𝑏) = (2nd ‘𝑏))
76	75	eqeq2d 2743	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝑎 = (𝐹‘𝑏) ↔ 𝑎 = (2nd ‘𝑏)))
77	76	rexbidva 3176	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏) ↔ ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
78	73, 77	mpbird 256	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
79	78	ralrimiva 3146	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
80		dffo3 7103	. 2 ⊢ (𝐹:𝐺–onto→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)))
81	5, 79, 80	sylanbrc 583	1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ⟨cop 4634   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   I cid 5573  dom cdm 5676  ran crn 5677   ↾ cres 5678  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  2^nd c2nd 7976   ≤ cle 11251  2c2 12269  ♯chash 14292  Vtxcvtx 28294  Edgcedg 28345  USPGraphcuspgr 28446  Pairscspr 46224

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-fz 13487  df-hash 14293  df-vtx 28296  df-iedg 28297  df-edg 28346  df-upgr 28380  df-uspgr 28448  df-spr 46225

This theorem is referenced by:  uspgrsprf1o  46606