Theorem uspgredgiedg 29153

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 29151	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
3		uspgredgiedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4		f1oeq3 6753	. . . . 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 7340	. . 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 2738	. . 3 ⊢ (𝐾 = (𝐼‘𝑥) ↔ (𝐼‘𝑥) = 𝐾)
10	9	reubii 3355	. 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 1541   ∈ wcel 2111  ∃!wreu 3344  dom cdm 5614  –1-1-onto→wf1o 6480  ‘cfv 6481  iEdgciedg 28975  Edgcedg 29025  USPGraphcuspgr 29126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-edg 29026  df-uspgr 29128

This theorem is referenced by: (None)