Theorem upgr1een 15968

Description: A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 15965 for a different way of specifying a graph with one edge. (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
upgr1een	⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)

Proof of Theorem upgr1een

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr1een.2o	. . 3 ⊢ (𝜑 → 𝐸 ≈ 2o)
2		en2 6993	. . 3 ⊢ (𝐸 ≈ 2o → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣})
4		eqid 2229	. . . . 5 ⊢ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)
5		upgr1een.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
6	5	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐾 ∈ 𝑋)
7		upgr1een.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉)
8	7	elpwid 3661	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ⊆ 𝑉)
9	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ⊆ 𝑉)
10		vex 2803	. . . . . . . . 9 ⊢ 𝑢 ∈ V
11	10	prid1 3775	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢, 𝑣}
12		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 = {𝑢, 𝑣})
13	11, 12	eleqtrrid 2319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝐸)
14	9, 13	sseldd 3226	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ 𝑉)
15		upgr1een.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑌)
16		opexg 4318	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → ⟨𝐾, 𝐸⟩ ∈ V)
17	5, 7, 16	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐾, 𝐸⟩ ∈ V)
18		snexg 4272	. . . . . . . . 9 ⊢ (⟨𝐾, 𝐸⟩ ∈ V → {⟨𝐾, 𝐸⟩} ∈ V)
19	17, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐾, 𝐸⟩} ∈ V)
20		opvtxfv 15866	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
21	15, 19, 20	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
22	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = 𝑉)
23	14, 22	eleqtrrd 2309	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
24		vex 2803	. . . . . . . . 9 ⊢ 𝑣 ∈ V
25	24	prid2 3776	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑢, 𝑣}
26	25, 12	eleqtrrid 2319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝐸)
27	9, 26	sseldd 3226	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ 𝑉)
28	27, 22	eleqtrrd 2309	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑣 ∈ (Vtx‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩))
29	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝐸 ≈ 2o)
30	12, 29	eqbrtrrd 4110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {𝑢, 𝑣} ≈ 2o)
31		pr2ne 7391	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑣 ∈ V) → ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣))
32	31	el2v 2806	. . . . . . . 8 ⊢ ({𝑢, 𝑣} ≈ 2o ↔ 𝑢 ≠ 𝑣)
33	30, 32	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → 𝑢 ≠ 𝑣)
34	33	olcd 739	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
35		dcne 2411	. . . . . 6 ⊢ (DECID 𝑢 = 𝑣 ↔ (𝑢 = 𝑣 ∨ 𝑢 ≠ 𝑣))
36	34, 35	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → DECID 𝑢 = 𝑣)
37		opiedgfv 15869	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ {⟨𝐾, 𝐸⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
38	15, 19, 37	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
39	38	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, 𝐸⟩})
40	12	opeq2d 3867	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝐾, 𝐸⟩ = ⟨𝐾, {𝑢, 𝑣}⟩)
41	40	sneqd 3680	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → {⟨𝐾, 𝐸⟩} = {⟨𝐾, {𝑢, 𝑣}⟩})
42	39, 41	eqtrd 2262	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → (iEdg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩) = {⟨𝐾, {𝑢, 𝑣}⟩})
43	4, 6, 23, 28, 36, 42	upgr1edc 15965	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = {𝑢, 𝑣}) → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)
44	43	ex 115	. . 3 ⊢ (𝜑 → (𝐸 = {𝑢, 𝑣} → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph))
45	44	exlimdvv 1944	. 2 ⊢ (𝜑 → (∃𝑢∃𝑣 𝐸 = {𝑢, 𝑣} → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph))
46	3, 45	mpd 13	1 ⊢ (𝜑 → ⟨𝑉, {⟨𝐾, 𝐸⟩}⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ≠ wne 2400  Vcvv 2800   ⊆ wss 3198  𝒫 cpw 3650  {csn 3667  {cpr 3668  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  2_oc2o 6571   ≈ cen 6902  Vtxcvtx 15856  iEdgciedg 15857  UPGraphcupgr 15935

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-upgren 15937

This theorem is referenced by:  umgr1een  15969  p1evtxdeqfilem  16122  p1evtxdeqfi  16123