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Mirrors > Home > MPE Home > Th. List > upgrle2 | Structured version Visualization version GIF version |
Description: An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.) |
Ref | Expression |
---|---|
upgrle2.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgrle2 | ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph) | |
2 | upgruhgr 28351 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
3 | upgrle2.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | uhgrfun 28315 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼) |
6 | 5 | funfnd 6576 | . . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼) |
7 | 6 | adantr 481 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼) |
8 | simpr 485 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼) | |
9 | eqid 2732 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
10 | 9, 3 | upgrle 28339 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) |
11 | 1, 7, 8, 10 | syl3anc 1371 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 dom cdm 5675 Fun wfun 6534 Fn wfn 6535 ‘cfv 6540 ≤ cle 11245 2c2 12263 ♯chash 14286 Vtxcvtx 28245 iEdgciedg 28246 UHGraphcuhgr 28305 UPGraphcupgr 28329 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-uhgr 28307 df-upgr 28331 |
This theorem is referenced by: upgr2pthnlp 28978 |
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