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Theorem uspgriedgedg 29194
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) → ∃!𝑘𝐸 𝑘 = (𝐼𝑋))
Distinct variable groups:   𝑘,𝐸   𝑘,𝐼   𝑘,𝑋
Allowed substitution hint:   𝐺(𝑘)

Proof of Theorem uspgriedgedg
StepHypRef Expression
1 uspgredgiedg.i . . . . . 6 𝐼 = (iEdg‘𝐺)
21uspgrf1oedg 29191 . . . . 5 (𝐺 ∈ USPGraph → 𝐼:dom 𝐼1-1-onto→(Edg‘𝐺))
3 f1of 6847 . . . . 5 (𝐼:dom 𝐼1-1-onto→(Edg‘𝐺) → 𝐼:dom 𝐼⟶(Edg‘𝐺))
42, 3syl 17 . . . 4 (𝐺 ∈ USPGraph → 𝐼:dom 𝐼⟶(Edg‘𝐺))
5 uspgredgiedg.e . . . . 5 𝐸 = (Edg‘𝐺)
6 feq3 6717 . . . . 5 (𝐸 = (Edg‘𝐺) → (𝐼:dom 𝐼𝐸𝐼:dom 𝐼⟶(Edg‘𝐺)))
75, 6ax-mp 5 . . . 4 (𝐼:dom 𝐼𝐸𝐼:dom 𝐼⟶(Edg‘𝐺))
84, 7sylibr 234 . . 3 (𝐺 ∈ USPGraph → 𝐼:dom 𝐼𝐸)
9 fdmeu 6964 . . 3 ((𝐼:dom 𝐼𝐸𝑋 ∈ dom 𝐼) → ∃!𝑘𝐸 (𝐼𝑋) = 𝑘)
108, 9sylan 580 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘𝐸 (𝐼𝑋) = 𝑘)
11 eqcom 2743 . . 3 (𝑘 = (𝐼𝑋) ↔ (𝐼𝑋) = 𝑘)
1211reubii 3388 . 2 (∃!𝑘𝐸 𝑘 = (𝐼𝑋) ↔ ∃!𝑘𝐸 (𝐼𝑋) = 𝑘)
1310, 12sylibr 234 1 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘𝐸 𝑘 = (𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!wreu 3377  dom cdm 5684  wf 6556  1-1-ontowf1o 6559  cfv 6560  iEdgciedg 29015  Edgcedg 29065  USPGraphcuspgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-edg 29066  df-uspgr 29168
This theorem is referenced by:  isuspgrim0  47877
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