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Theorem usgrafilem1 25940
 Description: The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)
Hypothesis
Ref Expression
usgrafis.f 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
Assertion
Ref Expression
usgrafilem1 dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
Distinct variable groups:   𝑥,𝐸   𝑥,𝑁
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem usgrafilem1
StepHypRef Expression
1 uncom 3719 . 2 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
2 rabxm 3915 . . 3 dom 𝐸 = ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸𝑥)})
3 df-nel 2783 . . . . . . 7 (𝑁 ∉ (𝐸𝑥) ↔ ¬ 𝑁 ∈ (𝐸𝑥))
43bicomi 213 . . . . . 6 𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∉ (𝐸𝑥))
54a1i 11 . . . . 5 (𝑥 ∈ dom 𝐸 → (¬ 𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∉ (𝐸𝑥)))
65rabbiia 3161 . . . 4 {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸𝑥)} = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}
76uneq2i 3726 . . 3 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸𝑥)}) = ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
82, 7eqtri 2632 . 2 dom 𝐸 = ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
9 usgrafis.f . . . . 5 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
109dmeqi 5247 . . . 4 dom 𝐹 = dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
11 ssrab2 3650 . . . . 5 {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸
12 ssdmres 5340 . . . . 5 ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸 ↔ dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
1311, 12mpbi 219 . . . 4 dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}
1410, 13eqtri 2632 . . 3 dom 𝐹 = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}
1514uneq1i 3725 . 2 (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
161, 8, 153eqtr4i 2642 1 dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900   ∪ cun 3538   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  ‘cfv 5804 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050 This theorem is referenced by:  usgrafilem2  25941  cusgrasizeinds  26004
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