Theorem upgredg 29115

Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)

Hypotheses

Ref	Expression
upgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
upgredg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
upgredg	⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Distinct variable groups:   𝐶,𝑎,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎,𝑏

Allowed substitution hints:   𝐸(𝑎,𝑏)

Proof of Theorem upgredg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
2		edgval 29027	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
4	1, 3	eqtrid 2778	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
5	4	eleq2d 2817	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ran (iEdg‘𝐺)))
6		upgredg.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2731	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgrf 29064	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
9	8	frnd 6659	. . . . 5 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
10	9	sseld 3928	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ ran (iEdg‘𝐺) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
11	5, 10	sylbid 240	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ 𝐸 → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
12	11	imp 406	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
13		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝐶 → (♯‘𝑥) = (♯‘𝐶))
14	13	breq1d 5099	. . . 4 ⊢ (𝑥 = 𝐶 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐶) ≤ 2))
15	14	elrab 3642	. . 3 ⊢ (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝐶) ≤ 2))
16		hashle2prv 14385	. . . 4 ⊢ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((♯‘𝐶) ≤ 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}))
17	16	biimpa 476	. . 3 ⊢ ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝐶) ≤ 2) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
18	15, 17	sylbi 217	. 2 ⊢ (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
19	12, 18	syl 17	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395   ∖ cdif 3894  ∅c0 4280  𝒫 cpw 4547  {csn 4573  {cpr 4575   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-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-fz 13408  df-hash 14238  df-edg 29026  df-upgr 29060

This theorem is referenced by:  upgrpredgv  29117  upgredg2vtx  29119  upgredgpr  29120  edglnl  29121  numedglnl  29122  isuspgrimlem  47934