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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 472 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph ) | |
2 | upgruhgr 25768 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) | |
3 | upgrle2.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | uhgrfun 25732 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼) |
6 | funfn 5833 | . . . 4 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
7 | 5, 6 | sylib 207 | . . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼) |
9 | simpr 476 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼) | |
10 | eqid 2610 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | 10, 3 | upgrle 25757 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) ≤ 2) |
12 | 1, 8, 9, 11 | syl3anc 1318 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) ≤ 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-uhgr 25724 df-upgr 25749 |
This theorem is referenced by: upgr2pthnlp 40938 |
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