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| Mirrors > Home > MPE Home > Th. List > uspgriedgedg | Structured version Visualization version GIF version | ||
| Description: In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| uspgredgiedg.e | ⊢ 𝐸 = (Edg‘𝐺) | 
| uspgredgiedg.i | ⊢ 𝐼 = (iEdg‘𝐺) | 
| Ref | Expression | 
|---|---|
| uspgriedgedg | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uspgredgiedg.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 2 | 1 | uspgrf1oedg 29191 | . . . . 5 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺)) | 
| 3 | f1of 6847 | . . . . 5 ⊢ (𝐼:dom 𝐼–1-1-onto→(Edg‘𝐺) → 𝐼:dom 𝐼⟶(Edg‘𝐺)) | |
| 4 | 2, 3 | syl 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 𝐼⟶𝐸 ↔ 𝐼:dom 𝐼⟶(Edg‘𝐺))) | |
| 7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (𝐼:dom 𝐼⟶𝐸 ↔ 𝐼:dom 𝐼⟶(Edg‘𝐺)) | 
| 8 | 4, 7 | sylibr 234 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼⟶𝐸) | 
| 9 | fdmeu 6964 | . . 3 ⊢ ((𝐼:dom 𝐼⟶𝐸 ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘) | |
| 10 | 8, 9 | sylan 580 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘) | 
| 11 | eqcom 2743 | . . 3 ⊢ (𝑘 = (𝐼‘𝑋) ↔ (𝐼‘𝑋) = 𝑘) | |
| 12 | 11 | reubii 3388 | . 2 ⊢ (∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋) ↔ ∃!𝑘 ∈ 𝐸 (𝐼‘𝑋) = 𝑘) | 
| 13 | 10, 12 | sylibr 234 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋)) | 
| 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-onto→wf1o 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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