Theorem uspgredgiedg 29378

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 29376	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺))
3		uspgredgiedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4		f1oeq3 6798	. . . . 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 236	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→𝐸)
7		f1ofveu 7392	. . 3 ⊢ ((𝐼:dom 𝐼–1-1-onto→𝐸 ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
8	6, 7	sylan 589	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
9		eqcom 2771	. . 3 ⊢ (𝐾 = (𝐼‘𝑥) ↔ (𝐼‘𝑥) = 𝐾)
10	9	reubii 3378	. 2 ⊢ (∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥) ↔ ∃!𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝐾)
11	8, 10	sylibr 236	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃!wreu 3367  dom cdm 5649  –1-1-onto→wf1o 6522  ‘cfv 6523  iEdgciedg 29200  Edgcedg 29250  USPGraphcuspgr 29351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-edg 29251  df-uspgr 29353

This theorem is referenced by: (None)