Theorem edgupgr 29169

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 29084	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . 4 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	2	eleq2d 2830	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran (iEdg‘𝐺)))
4		eqid 2740	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2740	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgrf 29121	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	6	frnd 6755	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
8	7	sseld 4007	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → 𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
9		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
10	9	breq1d 5176	. . . . . 6 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐸) ≤ 2))
11	10	elrab 3708	. . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2))
12		eldifsn 4811	. . . . . . . . 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 614	. . . . . . 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 1537   ∈ wcel 2108   ≠ wne 2946  {crab 3443   ∖ cdif 3973  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  dom cdm 5700  ran crn 5701  ‘cfv 6573   ≤ cle 11325  2c2 12348  ♯chash 14379  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082  UPGraphcupgr 29115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-edg 29083  df-upgr 29117

This theorem is referenced by:  upgrres1  29348  isuspgrim0  47756