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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 𝐸𝑁 ∉ (𝐸𝑥)})
Assertion
Ref Expression
cusgrares ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (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.
StepHypRef Expression
1 cusisusgra 25987 . . 3 (𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
2 cusgrares.f . . . 4 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
32usgrares1 25939 . . 3 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
41, 3sylan 487 . 2 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
5 iscusgra0 25986 . . . 4 (𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
6 usgraf1o 25887 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
7 f1ocnvdm 6440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸)
87adantll 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸)
9 elpri 4145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 ∈ {𝑛, 𝑘} → (𝑁 = 𝑛𝑁 = 𝑘))
10 vsnid 4156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑛 ∈ {𝑛}
11 sneq 4135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑁 = 𝑛 → {𝑁} = {𝑛})
1210, 11syl5eleqr 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 = 𝑛𝑛 ∈ {𝑁})
1312notnotd 137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 = 𝑛 → ¬ ¬ 𝑛 ∈ {𝑁})
14 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑛 ∉ {𝑁} ↔ ¬ 𝑛 ∈ {𝑁})
1513, 14sylnibr 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = 𝑛 → ¬ 𝑛 ∉ {𝑁})
16 vsnid 4156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑘 ∈ {𝑘}
17 sneq 4135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑁 = 𝑘 → {𝑁} = {𝑘})
1816, 17syl5eleqr 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑁 = 𝑘𝑘 ∈ {𝑁})
1918notnotd 137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 = 𝑘 → ¬ ¬ 𝑘 ∈ {𝑁})
20 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑘 ∉ {𝑁} ↔ ¬ 𝑘 ∈ {𝑁})
2119, 20sylnibr 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = 𝑘 → ¬ 𝑘 ∉ {𝑁})
2215, 21orim12i 537 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 = 𝑛𝑁 = 𝑘) → (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
239, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ {𝑛, 𝑘} → (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
24 ianor 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (¬ (𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ↔ (¬ 𝑛 ∉ {𝑁} ∨ ¬ 𝑘 ∉ {𝑁}))
2523, 24sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ {𝑛, 𝑘} → ¬ (𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}))
2625con2i 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → ¬ 𝑁 ∈ {𝑛, 𝑘})
27 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∉ {𝑛, 𝑘} ↔ ¬ 𝑁 ∈ {𝑛, 𝑘})
2826, 27sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → 𝑁 ∉ {𝑛, 𝑘})
2928ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → 𝑁 ∉ {𝑛, 𝑘})
30 f1ocnvfv2 6433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})
3130adantll 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-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}))
3331, 32syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑁 ∉ (𝐸‘(𝐸‘{𝑛, 𝑘})) ↔ 𝑁 ∉ {𝑛, 𝑘}))
3429, 33mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → 𝑁 ∉ (𝐸‘(𝐸‘{𝑛, 𝑘})))
35 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝐸‘{𝑛, 𝑘}) → (𝐸𝑥) = (𝐸‘(𝐸‘{𝑛, 𝑘})))
36 neleq2 2889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑥) = (𝐸‘(𝐸‘{𝑛, 𝑘})) → (𝑁 ∉ (𝐸𝑥) ↔ 𝑁 ∉ (𝐸‘(𝐸‘{𝑛, 𝑘}))))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝐸‘{𝑛, 𝑘}) → (𝑁 ∉ (𝐸𝑥) ↔ 𝑁 ∉ (𝐸‘(𝐸‘{𝑛, 𝑘}))))
3837elrab 3331 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ↔ ((𝐸‘{𝑛, 𝑘}) ∈ dom 𝐸𝑁 ∉ (𝐸‘(𝐸‘{𝑛, 𝑘}))))
398, 34, 38sylanbrc 695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
4039, 31jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
4140exp31 628 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ({𝑛, 𝑘} ∈ ran 𝐸 → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
4241com23 84 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∉ {𝑁} ∧ 𝑘 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
4342ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∉ {𝑁} → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
4443com14 94 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
456, 44syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 USGrph 𝐸 → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
4645ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) → (𝑘 ∉ {𝑁} → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘})))))
4746imp 444 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
4847adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))))
4948imp31 447 . . . . . . . . . . . . . . . . 17 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
50 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝐸‘{𝑛, 𝑘}) → (𝐸𝑦) = (𝐸‘(𝐸‘{𝑛, 𝑘})))
5150eqeq1d 2612 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐸‘{𝑛, 𝑘}) → ((𝐸𝑦) = {𝑛, 𝑘} ↔ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}))
5251rspcev 3282 . . . . . . . . . . . . . . . . 17 (((𝐸‘{𝑛, 𝑘}) ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∧ (𝐸‘(𝐸‘{𝑛, 𝑘})) = {𝑛, 𝑘}) → ∃𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) = {𝑛, 𝑘})
5349, 52syl 17 . . . . . . . . . . . . . . . 16 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ∃𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) = {𝑛, 𝑘})
54 usgrafun 25878 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸 → Fun 𝐸)
55 funfn 5833 . . . . . . . . . . . . . . . . . . 19 (Fun 𝐸𝐸 Fn dom 𝐸)
5654, 55sylib 207 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸𝐸 Fn dom 𝐸)
5756ad6antr 768 . . . . . . . . . . . . . . . . 17 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → 𝐸 Fn dom 𝐸)
58 ssrab2 3650 . . . . . . . . . . . . . . . . 17 {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸
59 fvelimab 6163 . . . . . . . . . . . . . . . . 17 ((𝐸 Fn dom 𝐸 ∧ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸) → ({𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ↔ ∃𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) = {𝑛, 𝑘}))
6057, 58, 59sylancl 693 . . . . . . . . . . . . . . . 16 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → ({𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ↔ ∃𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) = {𝑛, 𝑘}))
6153, 60mpbird 246 . . . . . . . . . . . . . . 15 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → {𝑛, 𝑘} ∈ (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}))
622rneqi 5273 . . . . . . . . . . . . . . . 16 ran 𝐹 = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
63 df-ima 5051 . . . . . . . . . . . . . . . 16 (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
6462, 63eqtr4i 2635 . . . . . . . . . . . . . . 15 ran 𝐹 = (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
6561, 64syl6eleqr 2699 . . . . . . . . . . . . . 14 (((((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) ∧ 𝑛 ∉ {𝑁}) → {𝑛, 𝑘} ∈ ran 𝐹)
6665exp31 628 . . . . . . . . . . . . 13 (((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹)))
6766ralimdva 2945 . . . . . . . . . . . 12 ((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑘})(𝑛 ∉ {𝑁} → {𝑛, 𝑘} ∈ ran 𝐹)))
6867imp 444 . . . . . . . . . . 11 (((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 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)
7068, 69sylib 207 . . . . . . . . . 10 (((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 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})
7271raleqi 3119 . . . . . . . . . 10 (∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 ↔ ∀𝑛 ∈ ((𝑉 ∖ {𝑘}) ∖ {𝑁}){𝑛, 𝑘} ∈ ran 𝐹)
7370, 72sylibr 223 . . . . . . . . 9 (((((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) ∧ 𝑘 ∉ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)
7473exp31 628 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) → (𝑘 ∉ {𝑁} → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
7574com23 84 . . . . . . 7 (((𝑉 USGrph 𝐸𝑁𝑉) ∧ 𝑘𝑉) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
7675ralimdva 2945 . . . . . 6 ((𝑉 USGrph 𝐸𝑁𝑉) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑘𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
7776impancom 455 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → (𝑁𝑉 → ∀𝑘𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
78 raldifb 3712 . . . . 5 (∀𝑘𝑉 (𝑘 ∉ {𝑁} → ∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹) ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)
7977, 78syl6ib 240 . . . 4 ((𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → (𝑁𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
805, 79syl 17 . . 3 (𝑉 ComplUSGrph 𝐸 → (𝑁𝑉 → ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
8180imp 444 . 2 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → ∀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 𝐸𝑁 ∉ (𝐸𝑥)}) ∈ V)
852, 84syl5eqel 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)))
8783, 85, 86syl2an 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)))
881, 82, 873syl 18 . . 3 (𝑉 ComplUSGrph 𝐸 → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
8988adantr 480 . 2 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → ((𝑉 ∖ {𝑁}) ComplUSGrph 𝐹 ↔ ((𝑉 ∖ {𝑁}) USGrph 𝐹 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∀𝑛 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹)))
904, 81, 89mpbir2and 959 1 ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)
 Colors of variables: wff setvar class 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
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