Theorem uspgrsprfo 41756

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 41754	. . 3 ⊢ 𝐹:𝐺⟶𝑃
5	4	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶𝑃)
6	1	eleq2i 2693	. . . . . . 7 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉))
7		selpw 4165	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
8	6, 7	bitri 264	. . . . . 6 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉))
9		eqidd 2623	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑉 = 𝑉)
10		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
11	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ∈ V)
12		f1oi 6174	. . . . . . . . . . . . . . . . 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 5457	. . . . . . . . . . . . . . . . 17 ⊢ dom ( I ↾ 𝑎) = 𝑎
15		f1oeq2 6128	. . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎)
18		sprvalpwle2 41739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
19	18	sseq2d 3633	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
20	19	biimpac 503	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
21	17, 20	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
22		f1oeq3 6129	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎))
23		sseq1 3626	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2} ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
24	22, 23	anbi12d 747	. . . . . . . . . . . . . 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	elabd 3352	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
26		resiexg 7102	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V → ( I ↾ 𝑎) ∈ V)
27	10, 26	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ( I ↾ 𝑎) ∈ V
28	27	f11o 7128	. . . . . . . . . . . . 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 224	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})
30	10	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V)
31	30	resiexd 6480	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V)
32	31	anim2i 593	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V))
33	32	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V))
34		isuspgrop 26056	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}))
36	29, 35	mpbird 247	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, ( I ↾ 𝑎)⟩ ∈ USPGraph )
37		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (Vtx‘𝑞) = (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩))
38	37	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉))
39		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (Edg‘𝑞) = (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩))
40	39	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
41	38, 40	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝑉, ( I ↾ 𝑎)⟩ → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)))
42	41	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)))
43		opvtxfv 25884	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
44	32, 43	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉)
45		edgopval 25944	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
46	32, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = ran ( I ↾ 𝑎))
47		rnresi 5479	. . . . . . . . . . . . . 14 ⊢ ran ( I ↾ 𝑎) = 𝑎
48	46, 47	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎)
49	44, 48	jca 554	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
50	49	ancoms 469	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ((Vtx‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑉 ∧ (Edg‘⟨𝑉, ( I ↾ 𝑎)⟩) = 𝑎))
51	36, 42, 50	rspcedvd 3317	. . . . . . . . . 10 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))
52	9, 51	jca 554	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
53	2	eleq2i 2693	. . . . . . . . . 10 ⊢ (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝑎⟩ ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
54	30	anim1i 592	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑎 ∈ V ∧ 𝑉 ∈ 𝑊))
55	54	ancomd 467	. . . . . . . . . . 11 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V))
56		eqeq1 2626	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑉 → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
57	56	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉))
58		eqeq2 2633	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉))
59		eqeq2 2633	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎))
60	58, 59	bi2anan9 917	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
61	60	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))
62	57, 61	anbi12d 747	. . . . . . . . . . . 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))))
63	62	opelopabga 4988	. . . . . . . . . . 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))))
64	55, 63	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))))
65	53, 64	syl5bb 272	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (⟨𝑉, 𝑎⟩ ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))))
66	52, 65	mpbird 247	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ⟨𝑉, 𝑎⟩ ∈ 𝐺)
67		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (2nd ‘𝑏) = (2nd ‘⟨𝑉, 𝑎⟩))
68	67	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑉, 𝑎⟩ → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
69	68	adantl 482	. . . . . . . 8 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑏 = ⟨𝑉, 𝑎⟩) → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩)))
70		op2ndg 7181	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
71	10, 70	mpan2 707	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑊 → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
72	71	adantl 482	. . . . . . . . 9 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (2nd ‘⟨𝑉, 𝑎⟩) = 𝑎)
73	72	eqcomd 2628	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 = (2nd ‘⟨𝑉, 𝑎⟩))
74	66, 69, 73	rspcedvd 3317	. . . . . . 7 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
75	74	ex 450	. . . . . 6 ⊢ (𝑎 ⊆ (Pairs‘𝑉) → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
76	8, 75	sylbi 207	. . . . 5 ⊢ (𝑎 ∈ 𝑃 → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
77	76	impcom 446	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))
78	1, 2, 3	uspgrsprfv 41753	. . . . . . 7 ⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏))
79	78	adantl 482	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝐹‘𝑏) = (2nd ‘𝑏))
80	79	eqeq2d 2632	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝑎 = (𝐹‘𝑏) ↔ 𝑎 = (2nd ‘𝑏)))
81	80	rexbidva 3049	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏) ↔ ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)))
82	77, 81	mpbird 247	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
83	82	ralrimiva 2966	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))
84		dffo3 6374	. 2 ⊢ (𝐹:𝐺–onto→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)))
85	5, 83, 84	sylanbrc 698	1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ⟨cop 4183   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   I cid 5023  dom cdm 5114  ran crn 5115   ↾ cres 5116  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  2^nd c2nd 7167   ≤ cle 10075  2c2 11070  #chash 13117  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-vtx 25876  df-iedg 25877  df-edg 25940  df-upgr 25977  df-uspgr 26045  df-spr 41728

This theorem is referenced by:  uspgrsprf1o  41757