Theorem edgupgr 29112

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 29027	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eleq2d 2817	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran (iEdg‘𝐺)))
4		eqid 2731	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2731	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgrf 29064	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	frnd 6659	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	7	sseld 3928	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → 𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
9		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
10	9	breq1d 5099	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐸) ≤ 2))
11	10	elrab 3642	. . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2))
12		eldifsn 4735	. . . . . . . . 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 1088	. . . . . . 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 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  {crab 3395   ∖ cdif 3894  ∅c0 4280  𝒫 cpw 4547  {csn 4573   class class class wbr 5089  dom cdm 5614  ran crn 5615  ‘cfv 6481   ≤ cle 11147  2c2 12180  ♯chash 14237  Vtxcvtx 28974  iEdgciedg 28975  Edgcedg 29025  UPGraphcupgr 29058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-edg 29026  df-upgr 29060

This theorem is referenced by:  upgrres1  29291  isuspgrim0  47933