Theorem edgupgr 29203

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 29118	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eleq2d 2822	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran (iEdg‘𝐺)))
4		eqid 2736	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2736	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgrf 29155	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	frnd 6676	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	7	sseld 3920	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → 𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
9		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
10	9	breq1d 5095	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐸) ≤ 2))
11	10	elrab 3634	. . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2))
12		eldifsn 4731	. . . . . . . . 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 616	. . . . . . 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 1542   ∈ wcel 2114   ≠ wne 2932  {crab 3389   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567   class class class wbr 5085  dom cdm 5631  ran crn 5632  ‘cfv 6498   ≤ cle 11180  2c2 12236  ♯chash 14292  Vtxcvtx 29065  iEdgciedg 29066  Edgcedg 29116  UPGraphcupgr 29149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-edg 29117  df-upgr 29151

This theorem is referenced by:  upgrres1  29382  isuspgrim0  48370