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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 26814 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
3 | upgrle2.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | uhgrfun 26778 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼) |
6 | 5 | funfnd 6379 | . . 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 2818 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
10 | 9, 3 | upgrle 26802 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) |
11 | 1, 7, 8, 10 | syl3anc 1363 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 ≤ cle 10664 2c2 11680 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 UHGraphcuhgr 26768 UPGraphcupgr 26792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-uhgr 26770 df-upgr 26794 |
This theorem is referenced by: upgr2pthnlp 27440 |
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