Theorem upgrle 25985

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
upgrle	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )

Proof of Theorem upgrle

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 |- V = (Vtx ` G )
2		isupgr.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	upgrfn 25982	. . . 4 |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
4	3	ffvelrnda 6359	. . 3 |- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
5	4	3impa 1259	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
6		fveq2 6191	. . . . 5 |- ( x = ( E ` F ) -> ( # ` x ) = ( # ` ( E ` F ) ) )
7	6	breq1d 4663	. . . 4 |- ( x = ( E ` F ) -> ( ( # ` x ) <_ 2 <-> ( # ` ( E ` F ) ) <_ 2 ) )
8	7	elrab 3363	. . 3 |- ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( ( E ` F ) e. ( ~P V \ { (/) } ) /\ ( # ` ( E ` F ) ) <_ 2 ) )
9	8	simprbi 480	. 2 |- ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ( # ` ( E ` F ) ) <_ 2 )
10	5, 9	syl 17	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   Fn wfn 5883   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   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-upgr 25977

This theorem is referenced by:  upgrfi  25986  upgrex  25987  upgrle2  26000  subupgr  26179  upgrewlkle2  26502