upgrle2 - Metamath Proof Explorer

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Theorem upgrle2 26000

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.)

**Hypothesis**
Ref	Expression
upgrle2.i	⊢ 𝐼 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
upgrle2	⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) ≤ 2)

**Proof of Theorem upgrle2**
Step	Hyp	Ref	Expression
1		simpl 473	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph )
2		upgruhgr 25997	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 25961	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6		funfn 5918	. . . 4 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
7	5, 6	sylib 208	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
8	7	adantr 481	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
9		simpr 477	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
10		eqid 2622	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11	10, 3	upgrle 25985	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) ≤ 2)
12	1, 8, 9, 11	syl3anc 1326	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼‘𝑋)) ≤ 2)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 ≤ cle 10075 2c2 11070 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 UPGraph cupgr 25975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-uhgr 25953 df-upgr 25977

This theorem is referenced by: upgr2pthnlp 26628

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