Theorem cusgrares 26001

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

Hypothesis

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

Assertion

Ref	Expression
cusgrares	⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)

Distinct variable groups:   𝑥,𝐸   𝑥,𝑁

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

Proof of Theorem cusgrares

Dummy variables 𝑦 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusisusgra 25987	. . 3 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)
2		cusgrares.f	. . . 4 ⊢ 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
3	2	usgrares1 25939	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
4	1, 3	sylan 487	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
5		iscusgra0 25986	. . . 4 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
6		usgraf1o 25887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
7		f1ocnvdm 6440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (◡𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸)
8	7	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (◡𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸)
9		elpri 4145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ {𝑛, 𝑘} → (𝑁 = 𝑛 ∨ 𝑁 = 𝑘))
10		vsnid 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ 𝑛 ∈ {𝑛}
11		sneq 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
12	10, 11	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 = 𝑛 → 𝑛 ∈ {𝑁})
13	12	notnotd 137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 = 𝑛 → ¬ ¬ 𝑛 ∈ {𝑁})
14		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑛 ∉ {𝑁} ↔ ¬ 𝑛 ∈ {𝑁})
15	13, 14	sylnibr 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 = 𝑛 → ¬ 𝑛 ∉ {𝑁})
16		vsnid 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ 𝑘 ∈ {𝑘}
17		sneq 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑁 = 𝑘 → {𝑁} = {𝑘})
18	16, 17	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 = 𝑘 → 𝑘 ∈ {𝑁})
19	18	notnotd 137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 = 𝑘 → ¬ ¬ 𝑘 ∈ {𝑁})
20		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∉ {𝑁} ↔ ¬ 𝑘 ∈ {𝑁})
21	19, 20	sylnibr 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 = 𝑘 → ¬ 𝑘 ∉ {𝑁})
22	15, 21	orim12i 537	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 = 𝑛 ∨ 𝑁 = 𝑘) → (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
23	9, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ {𝑛, 𝑘} → (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
24		ianor 508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ (𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ↔ (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
25	23, 24	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ {𝑛, 𝑘} → ¬ (𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}))
26	25	con2i 133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → ¬ 𝑁 ∈ {𝑛, 𝑘})
27		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑁 ∉ {𝑛, 𝑘} ↔ ¬ 𝑁 ∈ {𝑛, 𝑘})
28	26, 27	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → 𝑁 ∉ {𝑛, 𝑘})
29	28	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → 𝑁 ∉ {𝑛, 𝑘})
30		f1ocnvfv2 6433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})
31	30	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})
32		neleq2 2889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘} → (𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) ↔ 𝑁 ∉ {𝑛, 𝑘}))
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) ↔ 𝑁 ∉ {𝑛, 𝑘}))
34	29, 33	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → 𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘})))
35		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (◡𝐸‘{𝑛, 𝑘}) → (𝐸‘𝑥) = (𝐸‘(◡𝐸‘{𝑛, 𝑘})))
36		neleq2 2889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐸‘𝑥) = (𝐸‘(◡𝐸‘{𝑛, 𝑘})) → (𝑁 ∉ (𝐸‘𝑥) ↔ 𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘}))))
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (◡𝐸‘{𝑛, 𝑘}) → (𝑁 ∉ (𝐸‘𝑥) ↔ 𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘}))))
38	37	elrab 3331	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ↔ ((◡𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸 ∧ 𝑁 ∉ (𝐸‘(◡𝐸‘{𝑛, 𝑘}))))
39	8, 34, 38	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
40	39, 31	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸–1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
41	40	exp31 628	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ({𝑛, 𝑘} ∈ ran 𝐸 → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
42	41	com23 84	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
43	42	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∉ {𝑁} → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
44	43	com14 94	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
45	6, 44	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 USGrph 𝐸 → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
46	45	ad2antrr 758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
47	46	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
48	47	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
49	48	imp31 447	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
50		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (◡𝐸‘{𝑛, 𝑘}) → (𝐸‘𝑦) = (𝐸‘(◡𝐸‘{𝑛, 𝑘})))
51	50	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (◡𝐸‘{𝑛, 𝑘}) → ((𝐸‘𝑦) = {𝑛, 𝑘} ↔ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
52	51	rspcev 3282	. . . . . . . . . . . . . . . . 17 ⊢ (((◡𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∧ (𝐸‘(◡𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}) → ∃𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) = {𝑛, 𝑘})
53	49, 52	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ∃𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) = {𝑛, 𝑘})
54		usgrafun 25878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
55		funfn 5833	. . . . . . . . . . . . . . . . . . 19 ⊢ (Fun 𝐸 ↔ 𝐸 Fn dom 𝐸)
56	54, 55	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 USGrph 𝐸 → 𝐸 Fn dom 𝐸)
57	56	ad6antr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → 𝐸 Fn dom 𝐸)
58		ssrab2 3650	. . . . . . . . . . . . . . . . 17 ⊢ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸
59		fvelimab 6163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 Fn dom 𝐸 ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸) → ({𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ↔ ∃𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) = {𝑛, 𝑘}))
60	57, 58, 59	sylancl 693	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ↔ ∃𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} (𝐸‘𝑦) = {𝑛, 𝑘}))
61	53, 60	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → {𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}))
62	2	rneqi 5273	. . . . . . . . . . . . . . . 16 ⊢ ran 𝐹 = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
63		df-ima 5051	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
64	62, 63	eqtr4i 2635	. . . . . . . . . . . . . . 15 ⊢ ran 𝐹 = (𝐸 “ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
65	61, 64	syl6eleqr 2699	. . . . . . . . . . . . . 14 ⊢ (((((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → {𝑛, 𝑘} ∈ ran 𝐹)
66	65	exp31 628	. . . . . . . . . . . . 13 ⊢ (((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹)))
67	66	ralimdva 2945	. . . . . . . . . . . 12 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑘})(𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹)))
68	67	imp 444	. . . . . . . . . . 11 ⊢ (((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑛 ∈ (𝑉 ∖ {𝑘})(𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹))
69		raldifb 3712	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑘})(𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹) ↔ ∀𝑛 ∈ ((𝑉 ∖ {𝑘}) ∖ {𝑁}){𝑛, 𝑘} ∈ ran 𝐹)
70	68, 69	sylib 207	. . . . . . . . . 10 ⊢ (((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑛 ∈ ((𝑉 ∖ {𝑘}) ∖ {𝑁}){𝑛, 𝑘} ∈ ran 𝐹)
71		dif32 3850	. . . . . . . . . . 11 ⊢ ((𝑉 ∖ {𝑁}) ∖ {𝑘}) = ((𝑉 ∖ {𝑘}) ∖ {𝑁})
72	71	raleqi 3119	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 ↔ ∀𝑛 ∈ ((𝑉 ∖ {𝑘}) ∖ {𝑁}){𝑛, 𝑘} ∈ ran 𝐹)
73	70, 72	sylibr 223	. . . . . . . . 9 ⊢ (((((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)
74	73	exp31 628	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) → (𝑘 ∉ {𝑁} → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
75	74	com23 84	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 𝑘 ∈ 𝑉) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
76	75	ralimdva 2945	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑘 ∈ 𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
77	76	impancom 455	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → ∀𝑘 ∈ 𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
78		raldifb 3712	. . . . 5 ⊢ (∀𝑘 ∈ 𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹) ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)
79	77, 78	syl6ib 240	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
80	5, 79	syl 17	. . 3 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑁 ∈ 𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
81	80	imp 444	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)
82		usgrav 25867	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
83		difexg 4735	. . . . 5 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
84		resexg 5362	. . . . . 6 ⊢ (𝐸 ∈ V → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) ∈ V)
85	2, 84	syl5eqel 2692	. . . . 5 ⊢ (𝐸 ∈ V → 𝐹 ∈ V)
86		iscusgra 25985	. . . . 5 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ 𝐹 ∈ V) → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
87	83, 85, 86	syl2an 493	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
88	1, 82, 87	3syl 18	. . 3 ⊢ (𝑉 ComplUSGrph 𝐸 → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
89	88	adantr 480	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
90	4, 81, 89	mpbir2and 959	1 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804   USGrph cusg 25859   ComplUSGrph ccusgra 25947

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  df-cusgra 25950

This theorem is referenced by:  cusgrasizeinds  26004  cusgrasize  26006