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 e. UPGraph /\ X e. dom I ) -> ( # ` ( I ` X ) ) <_ 2 )

Proof of Theorem upgrle2

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( G e. UPGraph /\ X e. dom I ) -> G e. UPGraph )
2		upgruhgr 25997	. . . . 5 |- ( G e. UPGraph -> G e. UHGraph )
3		upgrle2.i	. . . . . 6 |- I = (iEdg ` G )
4	3	uhgrfun 25961	. . . . 5 |- ( G e. UHGraph -> Fun I )
5	2, 4	syl 17	. . . 4 |- ( G e. UPGraph -> Fun I )
6		funfn 5918	. . . 4 |- ( Fun I <-> I Fn dom I )
7	5, 6	sylib 208	. . 3 |- ( G e. UPGraph -> I Fn dom I )
8	7	adantr 481	. 2 |- ( ( G e. UPGraph /\ X e. dom I ) -> I Fn dom I )
9		simpr 477	. 2 |- ( ( G e. UPGraph /\ X e. dom I ) -> X e. dom I )
10		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
11	10, 3	upgrle 25985	. 2 |- ( ( G e. UPGraph /\ I Fn dom I /\ X e. dom I ) -> ( # ` ( I ` X ) ) <_ 2 )
12	1, 8, 9, 11	syl3anc 1326	1 |- ( ( G e. UPGraph /\ X e. dom I ) -> ( # ` ( I ` X ) ) <_ 2 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. 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