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| Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) | 
| Ref | Expression | 
|---|---|
| isupgr.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| isupgr.e | ⊢ 𝐸 = (iEdg‘𝐺) | 
| Ref | Expression | 
|---|---|
| upgrle | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isupgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | isupgr.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 3 | 1, 2 | upgrfn 29105 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 4 | 3 | ffvelcdmda 7103 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 5 | 4 | 3impa 1109 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 6 | fveq2 6905 | . . . . 5 ⊢ (𝑥 = (𝐸‘𝐹) → (♯‘𝑥) = (♯‘(𝐸‘𝐹))) | |
| 7 | 6 | breq1d 5152 | . . . 4 ⊢ (𝑥 = (𝐸‘𝐹) → ((♯‘𝑥) ≤ 2 ↔ (♯‘(𝐸‘𝐹)) ≤ 2)) | 
| 8 | 7 | elrab 3691 | . . 3 ⊢ ((𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ ((𝐸‘𝐹) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝐹)) ≤ 2)) | 
| 9 | 8 | simprbi 496 | . 2 ⊢ ((𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘(𝐸‘𝐹)) ≤ 2) | 
| 10 | 5, 9 | syl 17 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 {csn 4625 class class class wbr 5142 Fn wfn 6555 ‘cfv 6560 ≤ cle 11297 2c2 12322 ♯chash 14370 Vtxcvtx 29014 iEdgciedg 29015 UPGraphcupgr 29098 | 
| 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-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 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-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-fv 6568 df-upgr 29100 | 
| This theorem is referenced by: upgrfi 29109 upgrex 29110 upgrle2 29123 subupgr 29305 upgrewlkle2 29625 | 
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