Theorem isusgropen 15920

Description: The property of being an undirected simple graph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 30-Nov-2020.)

Assertion

Ref	Expression
isusgropen	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ 𝒫 𝑉 ∣ 𝑝 ≈ 2o}))

Distinct variable groups:   𝐸,𝑝   𝑉,𝑝   𝑊,𝑝   𝑋,𝑝

Proof of Theorem isusgropen

Step	Hyp	Ref	Expression
1		opexg 4291	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ⟨𝑉, 𝐸⟩ ∈ V)
2		eqid 2207	. . . 4 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3		eqid 2207	. . . 4 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
4	2, 3	isusgren 15913	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∣ 𝑝 ≈ 2o}))
5	1, 4	syl 14	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∣ 𝑝 ≈ 2o}))
6		opiedgfv 15785	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
7	6	dmeqd 4900	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8		opvtxfv 15782	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
9	8	pweqd 3632	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
10	9	rabeqdv 2771	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∣ 𝑝 ≈ 2o} = {𝑝 ∈ 𝒫 𝑉 ∣ 𝑝 ≈ 2o})
11	6, 7, 10	f1eq123d 5537	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ((iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∣ 𝑝 ≈ 2o} ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ 𝒫 𝑉 ∣ 𝑝 ≈ 2o}))
12	5, 11	bitrd 188	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ 𝒫 𝑉 ∣ 𝑝 ≈ 2o}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2178  {crab 2490  Vcvv 2777  𝒫 cpw 3627  ⟨cop 3647   class class class wbr 4060  dom cdm 4694  –1-1→wf1 5288  ‘cfv 5291  2_oc2o 6521   ≈ cen 6850  Vtxcvtx 15772  iEdgciedg 15773  USGraphcusgr 15909

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-cnre 8073

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-sub 8282  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-5 9135  df-6 9136  df-7 9137  df-8 9138  df-9 9139  df-n0 9333  df-dec 9542  df-ndx 12996  df-slot 12997  df-base 12999  df-edgf 15765  df-vtx 15774  df-iedg 15775  df-usgren 15911

This theorem is referenced by:  usgrop  15921