Theorem uspgredgiedg 29155

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

Hypotheses

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

Assertion

Ref	Expression
uspgredgiedg	⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼   𝑥,𝐾

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem uspgredgiedg

Step	Hyp	Ref	Expression
1		uspgredgiedg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	uspgrf1oedg 29153	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
3		uspgredgiedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4		f1oeq3 6772	. . . . 5 ⊢ (𝐸 = (Edg‘𝐺) → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
6	2, 5	sylibr 234	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)
7		f1ofveu 7363	. . 3 ⊢ ((𝐼:dom 𝐼–1-1-onto→𝐸 ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
8	6, 7	sylan 580	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
9		eqcom 2736	. . 3 ⊢ (𝐾 = (𝐼‘𝑥) ↔ (𝐼‘𝑥) = 𝐾)
10	9	reubii 3360	. 2 ⊢ (∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥) ↔ ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
11	8, 10	sylibr 234	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!wreu 3349  dom cdm 5631  –1-1-onto→wf1o 6498  ‘cfv 6499  iEdgciedg 28977  Edgcedg 29027  USPGraphcuspgr 29128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-edg 29028  df-uspgr 29130

This theorem is referenced by: (None)