Theorem edgupgr 29151

Description: Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)

Assertion

Ref	Expression
edgupgr	⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))

Proof of Theorem edgupgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		edgval 29066	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eleq2d 2827	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran (iEdg‘𝐺)))
4		eqid 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2737	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgrf 29103	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	frnd 6744	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	7	sseld 3982	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → 𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
9		fveq2 6906	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
10	9	breq1d 5153	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐸) ≤ 2))
11	10	elrab 3692	. . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2))
12		eldifsn 4786	. . . . . . . . 9 ⊢ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅))
13	12	biimpi 216	. . . . . . . 8 ⊢ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅))
14	13	anim1i 615	. . . . . . 7 ⊢ ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅) ∧ (♯‘𝐸) ≤ 2))
15		df-3an 1089	. . . . . . 7 ⊢ ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2) ↔ ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅) ∧ (♯‘𝐸) ≤ 2))
16	14, 15	sylibr 234	. . . . . 6 ⊢ ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))
17	16	a1i 11	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
18	11, 17	biimtrid 242	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
19	8, 18	syld 47	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
20	3, 19	sylbid 240	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
21	20	imp 406	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  {crab 3436   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  dom cdm 5685  ran crn 5686  ‘cfv 6561   ≤ cle 11296  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UPGraphcupgr 29097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29065  df-upgr 29099

This theorem is referenced by:  upgrres1  29330  isuspgrim0  47872