Theorem uspgriedgedg 29208

Description: In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.)

Hypotheses

Ref	Expression
uspgredgiedg.e	⊢ 𝐸 = (Edg‘𝐺)
uspgredgiedg.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
uspgriedgedg	⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋))

Distinct variable groups:   𝑘,𝐸   𝑘,𝐼   𝑘,𝑋

Allowed substitution hint:   𝐺(𝑘)

Proof of Theorem uspgriedgedg

Step	Hyp	Ref	Expression
1		uspgredgiedg.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	uspgrf1oedg 29205	. . . . 5 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
3		f1of 6849	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺) → 𝐼:dom 𝐼⟶(Edg‘𝐺))
4	2, 3	syl 17	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼⟶(Edg‘𝐺))
5		uspgredgiedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6		feq3 6719	. . . . 5 ⊢ (𝐸 = (Edg‘𝐺) → (𝐼:dom 𝐼⟶𝐸 ↔ 𝐼:dom 𝐼⟶(Edg‘𝐺)))
7	5, 6	ax-mp 5	. . . 4 ⊢ (𝐼:dom 𝐼⟶𝐸 ↔ 𝐼:dom 𝐼⟶(Edg‘𝐺))
8	4, 7	sylibr 234	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼⟶𝐸)
9		fdmeu 6965	. . 3 ⊢ ((𝐼:dom 𝐼⟶𝐸 ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘)
10	8, 9	sylan 580	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘)
11		eqcom 2742	. . 3 ⊢ (𝑘 = (𝐼‘𝑋) ↔ (𝐼‘𝑋) = 𝑘)
12	11	reubii 3387	. 2 ⊢ (∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋) ↔ ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘)
13	10, 12	sylibr 234	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃!wreu 3376  dom cdm 5689  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  iEdgciedg 29029  Edgcedg 29079  USPGraphcuspgr 29180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-edg 29080  df-uspgr 29182

This theorem is referenced by:  isuspgrim0  47810