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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})

Assertion

Ref	Expression
usgrafilem1	⊢ dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})

Distinct variable groups:   𝑥,𝐸   𝑥,𝑁

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem usgrafilem1

Step	Hyp	Ref	Expression
1		uncom 3719	. 2 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})
2		rabxm 3915	. . 3 ⊢ dom 𝐸 = ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸‘𝑥)})
3		df-nel 2783	. . . . . . 7 ⊢ (𝑁 ∉ (𝐸‘𝑥) ↔ ¬ 𝑁 ∈ (𝐸‘𝑥))
4	3	bicomi 213	. . . . . 6 ⊢ (¬ 𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∉ (𝐸‘𝑥))
5	4	a1i 11	. . . . 5 ⊢ (𝑥 ∈ dom 𝐸 → (¬ 𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∉ (𝐸‘𝑥)))
6	5	rabbiia 3161	. . . 4 ⊢ {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸‘𝑥)} = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}
7	6	uneq2i 3726	. . 3 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ ¬ 𝑁 ∈ (𝐸‘𝑥)}) = ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
8	2, 7	eqtri 2632	. 2 ⊢ dom 𝐸 = ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
9		usgrafis.f	. . . . 5 ⊢ 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
10	9	dmeqi 5247	. . . 4 ⊢ dom 𝐹 = dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
11		ssrab2 3650	. . . . 5 ⊢ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸
12		ssdmres 5340	. . . . 5 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ⊆ dom 𝐸 ↔ dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})
13	11, 12	mpbi 219	. . . 4 ⊢ dom (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}) = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}
14	10, 13	eqtri 2632	. . 3 ⊢ dom 𝐹 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)}
15	14	uneq1i 3725	. 2 ⊢ (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)} ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})
16	1, 8, 15	3eqtr4i 2642	1 ⊢ dom 𝐸 = (dom 𝐹 ∪ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})

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