Theorem uspgredgiedg 29252

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 29250	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
3		uspgredgiedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4		f1oeq3 6765	. . . . 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 7354	. . 3 ⊢ ((𝐼:dom 𝐼–1-1-onto→𝐸 ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
8	6, 7	sylan 581	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
9		eqcom 2744	. . 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 1542   ∈ wcel 2114  ∃!wreu 3349  dom cdm 5625  –1-1-onto→wf1o 6492  ‘cfv 6493  iEdgciedg 29074  Edgcedg 29124  USPGraphcuspgr 29225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-edg 29125  df-uspgr 29227

This theorem is referenced by: (None)