Theorem umgr1een 16137

Description: A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)

Hypotheses

Ref	Expression
upgr1een.k	⊢ (𝜑 → 𝐾 ∈ 𝑋)
upgr1een.v	⊢ (𝜑 → 𝑉 ∈ 𝑌)
upgr1een.e	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
upgr1een.2o	⊢ (𝜑 → 𝐸 ≈ 2o)

Assertion

Ref	Expression
umgr1een	⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph)

Proof of Theorem umgr1een

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr1een.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
2		breq1 4114	. . . . 5 ⊢ (𝑥 = 𝐸 → (𝑥 ≈ 2o ↔ 𝐸 ≈ 2o))
3		upgr1een.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
4		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
5		opexg 4346	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
6	1, 3, 5	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
7		snexg 4299	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
9		opvtxfv 16034	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
10	4, 8, 9	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
11	10	pweqd 3676	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝒫 𝑉)
12	3, 11	eleqtrrd 2314	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
13		upgr1een.2o	. . . . 5 ⊢ (𝜑 → 𝐸 ≈ 2o)
14	2, 12, 13	elrabd 2977	. . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o})
15	1, 14	fsnd 5661	. . 3 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩}:{𝐾}⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o})
16		opiedgfv 16037	. . . . 5 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
17	4, 8, 16	syl2anc 411	. . . 4 ⊢ (𝜑 → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
18	17	dmeqd 4960	. . . . 5 ⊢ (𝜑 → dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = dom {⟨𝐾, 𝐸⟩})
19		dmsnopg 5236	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑉 → dom {⟨𝐾, 𝐸⟩} = {𝐾})
20	3, 19	syl 14	. . . . 5 ⊢ (𝜑 → dom {⟨𝐾, 𝐸⟩} = {𝐾})
21	18, 20	eqtrd 2267	. . . 4 ⊢ (𝜑 → dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {𝐾})
22	17, 21	feq12d 5500	. . 3 ⊢ (𝜑 → ((iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩):dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o} ↔ {⟨𝐾, 𝐸⟩}:{𝐾}⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o}))
23	15, 22	mpbird 167	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩):dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o})
24	1, 4, 3, 13	upgr1een 16136	. . 3 ⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
25		eqid 2234	. . . 4 ⊢ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
26		eqid 2234	. . . 4 ⊢ (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
27	25, 26	isumgren 16117	. . 3 ⊢ (⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph → (⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph ↔ (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩):dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o}))
28	24, 27	syl 14	. 2 ⊢ (𝜑 → (⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph ↔ (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩):dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o}))
29	23, 28	mpbird 167	1 ⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UMGraph)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {crab 2526  Vcvv 2815  𝒫 cpw 3671  {csn 3691  ⟨cop 3694   class class class wbr 4111  dom cdm 4751  ⟶wf 5350  ‘cfv 5354  2_oc2o 6643   ≈ cen 6975  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103  UMGraphcumgr 16104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-upgren 16105  df-umgren 16106

This theorem is referenced by:  p1evtxdp1fi  16325