Theorem umgr1een 16003

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 4091	. . . . 5 ⊢ (𝑥 = 𝐸 → (𝑥 ≈ 2o ↔ 𝐸 ≈ 2o))
3		upgr1een.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
4		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
5		opexg 4320	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
6	1, 3, 5	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
7		snexg 4274	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
9		opvtxfv 15900	. . . . . . . 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 3657	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝒫 𝑉)
12	3, 11	eleqtrrd 2311	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
13		upgr1een.2o	. . . . 5 ⊢ (𝜑 → 𝐸 ≈ 2o)
14	2, 12, 13	elrabd 2964	. . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o})
15	1, 14	fsnd 5629	. . 3 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩}:{𝐾}⟶{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) ∣ 𝑥 ≈ 2o})
16		opiedgfv 15903	. . . . 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 4933	. . . . 5 ⊢ (𝜑 → dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = dom {⟨𝐾, 𝐸⟩})
19		dmsnopg 5208	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑉 → dom {⟨𝐾, 𝐸⟩} = {𝐾})
20	3, 19	syl 14	. . . . 5 ⊢ (𝜑 → dom {⟨𝐾, 𝐸⟩} = {𝐾})
21	18, 20	eqtrd 2264	. . . 4 ⊢ (𝜑 → dom (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {𝐾})
22	17, 21	feq12d 5472	. . 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 16002	. . 3 ⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
25		eqid 2231	. . . 4 ⊢ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
26		eqid 2231	. . . 4 ⊢ (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
27	25, 26	isumgren 15983	. . 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 1397   ∈ wcel 2202  {crab 2514  Vcvv 2802  𝒫 cpw 3652  {csn 3669  ⟨cop 3672   class class class wbr 4088  dom cdm 4725  ⟶wf 5322  ‘cfv 5326  2_oc2o 6579   ≈ cen 6910  Vtxcvtx 15890  iEdgciedg 15891  UPGraphcupgr 15969  UMGraphcumgr 15970

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-cnre 8146

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-1o 6585  df-2o 6586  df-er 6705  df-en 6913  df-sub 8355  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-5 9208  df-6 9209  df-7 9210  df-8 9211  df-9 9212  df-n0 9406  df-dec 9615  df-ndx 13106  df-slot 13107  df-base 13109  df-edgf 15883  df-vtx 15892  df-iedg 15893  df-upgren 15971  df-umgren 15972

This theorem is referenced by:  p1evtxdp1fi  16191