Theorem usgrares1 25939

Description: Restricting an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)

Hypothesis

Ref	Expression
usgrafis.f	⊢ 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})

Assertion

Ref	Expression
usgrares1	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)

Distinct variable groups:   𝑥,𝐸   𝑥,𝑁

Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem usgrares1

Dummy variables 𝑦 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgraf1 25889	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→ran 𝐸)
2	1	adantr 480	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → 𝐸:dom 𝐸–1-1→ran 𝐸)
3		ssrab2 3650	. . . . 5 ⊢ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸
4		f1ssres 6021	. . . . 5 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}):{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}–1-1→ran 𝐸)
5	2, 3, 4	sylancl 693	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}):{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}–1-1→ran 𝐸)
6		usgrafis.f	. . . . . 6 ⊢ 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
7	6	a1i 11	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}))
8	6	dmeqi 5247	. . . . . 6 ⊢ dom 𝐹 = dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
9	3	a1i 11	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸)
10		ssdmres 5340	. . . . . . 7 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸 ↔ dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
11	9, 10	sylib 207	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
12	8, 11	syl5eq 2656	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → dom 𝐹 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
13		eqidd 2611	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ran 𝐸 = ran 𝐸)
14	7, 12, 13	f1eq123d 6044	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝐹:dom 𝐹–1-1→ran 𝐸 ↔ (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}):{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}–1-1→ran 𝐸))
15	5, 14	mpbird 246	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → 𝐹:dom 𝐹–1-1→ran 𝐸)
16	6	rneqi 5273	. . . . 5 ⊢ ran 𝐹 = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
17		df-ima 5051	. . . . 5 ⊢ (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
18	16, 17	eqtr4i 2635	. . . 4 ⊢ ran 𝐹 = (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
19		eqidd 2611	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑁 = 𝑁)
20		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐸‘𝑥) = (𝐸‘𝑦))
21	19, 20	neleq12d 2887	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁 ∉ (𝐸‘𝑥) ↔ 𝑁 ∉ (𝐸‘𝑦)))
22	21	elrab 3331	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ↔ (𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)))
23		usgraf0 25877	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
24		f1f 6014	. . . . . . . . . . . . . . 15 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
25		ffvelrn 6265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑦 ∈ dom 𝐸) → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
26		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = (𝐸‘𝑦) → (#‘𝑒) = (#‘(𝐸‘𝑦)))
27	26	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = (𝐸‘𝑦) → ((#‘𝑒) = 2 ↔ (#‘(𝐸‘𝑦)) = 2))
28	27	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ↔ ((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2))
29		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∉ (𝐸‘𝑦) ↔ ¬ 𝑁 ∈ (𝐸‘𝑦))
30	29	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∉ (𝐸‘𝑦) → ¬ 𝑁 ∈ (𝐸‘𝑦))
31	30	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → ¬ 𝑁 ∈ (𝐸‘𝑦))
32		disjsn 4192	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸‘𝑦) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝐸‘𝑦))
33	31, 32	sylibr 223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → ((𝐸‘𝑦) ∩ {𝑁}) = ∅)
34		elpwi 4117	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐸‘𝑦) ∈ 𝒫 𝑉 → (𝐸‘𝑦) ⊆ 𝑉)
35	34	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑦) ⊆ 𝑉)
36		reldisj 3972	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸‘𝑦) ⊆ 𝑉 → (((𝐸‘𝑦) ∩ {𝑁}) = ∅ ↔ (𝐸‘𝑦) ⊆ (𝑉 ∖ {𝑁})))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → (((𝐸‘𝑦) ∩ {𝑁}) = ∅ ↔ (𝐸‘𝑦) ⊆ (𝑉 ∖ {𝑁})))
38	33, 37	mpbid 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑦) ⊆ (𝑉 ∖ {𝑁}))
39		fvex 6113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸‘𝑦) ∈ V
40	39	elpw 4114	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐸‘𝑦) ⊆ (𝑉 ∖ {𝑁}))
41	38, 40	sylibr 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → (𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}))
42		simpllr 795	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → (#‘(𝐸‘𝑦)) = 2)
43	41, 42	jca 553	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) ∧ 𝑁 ∉ (𝐸‘𝑦)) ∧ 𝑁 ∈ 𝑉) → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))
44	43	exp31 628	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸‘𝑦)) = 2) → (𝑁 ∉ (𝐸‘𝑦) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))))
45	28, 44	sylbi 206	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁 ∉ (𝐸‘𝑦) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))))
46	25, 45	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑦 ∈ dom 𝐸) → (𝑁 ∉ (𝐸‘𝑦) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))))
47	46	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑦 ∈ dom 𝐸 → (𝑁 ∉ (𝐸‘𝑦) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2)))))
48	24, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑦 ∈ dom 𝐸 → (𝑁 ∉ (𝐸‘𝑦) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2)))))
49	48	impd 446	. . . . . . . . . . . . 13 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → ((𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)) → (𝑁 ∈ 𝑉 → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))))
50	49	com23 84	. . . . . . . . . . . 12 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁 ∈ 𝑉 → ((𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)) → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))))
51	50	imp31 447	. . . . . . . . . . 11 ⊢ (((𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦))) → ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))
52	27	elrab 3331	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ((𝐸‘𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸‘𝑦)) = 2))
53	51, 52	sylibr 223	. . . . . . . . . 10 ⊢ (((𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦))) → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
54	53	exp31 628	. . . . . . . . 9 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁 ∈ 𝑉 → ((𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)) → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})))
55	23, 54	syl 17	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ 𝑉 → ((𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)) → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})))
56	55	imp 444	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝑦 ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘𝑦)) → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
57	22, 56	syl5bi 231	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} → (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
58	57	ralrimiv 2948	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
59		usgrafun 25878	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
60	3	a1i 11	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸)
61	59, 60	jca 553	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸))
62	61	adantr 480	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸))
63		funimass4 6157	. . . . . 6 ⊢ ((Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸) → ((𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ∀𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
64	62, 63	syl 17	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ∀𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
65	58, 64	mpbird 246	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
66	18, 65	syl5eqss 3612	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ran 𝐹 ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
67		f1ssr 6020	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐸 ∧ ran 𝐹 ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}) → 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
68	15, 66, 67	syl2anc 691	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
69		usgrav 25867	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
70		difexg 4735	. . . . 5 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
71		resexg 5362	. . . . . 6 ⊢ (𝐸 ∈ V → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ∈ V)
72	6, 71	syl5eqel 2692	. . . . 5 ⊢ (𝐸 ∈ V → 𝐹 ∈ V)
73		isusgra0 25876	. . . . 5 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ 𝐹 ∈ V) → ((𝑉 ∖ {𝑁}) USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
74	70, 72, 73	syl2an 493	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 ∖ {𝑁}) USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
75	69, 74	syl 17	. . 3 ⊢ (𝑉 USGrph 𝐸 → ((𝑉 ∖ {𝑁}) USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
76	75	adantr 480	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝑉 ∖ {𝑁}) USGrph 𝐹 ↔ 𝐹:dom 𝐹–1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
77	68, 76	mpbird 246	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862

This theorem is referenced by:  usgrafis  25944  cusgrares  26001