Theorem lfgredgge2 26019

Description: An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.)

Hypotheses

Ref	Expression
lfuhgrnloopv.i	|- I = (iEdg ` G )
lfuhgrnloopv.a	|- A = dom I
lfuhgrnloopv.e	|- E = { x e. ~P V | 2 <_ ( # ` x ) }

Assertion

Ref	Expression
lfgredgge2	|- ( ( I : A --> E /\ X e. A ) -> 2 <_ ( # ` ( I ` X ) ) )

Distinct variable groups:   x, A   x, I   x, V

Allowed substitution hints:   E( x)   G( x)   X( x)

Proof of Theorem lfgredgge2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- A = A
2		lfuhgrnloopv.e	. . . . 5 |- E = { x e. ~P V | 2 <_ ( # ` x ) }
3	1, 2	feq23i 6039	. . . 4 |- ( I : A --> E <-> I : A --> { x e. ~P V | 2 <_ ( # ` x ) } )
4	3	biimpi 206	. . 3 |- ( I : A --> E -> I : A --> { x e. ~P V | 2 <_ ( # ` x ) } )
5	4	ffvelrnda 6359	. 2 |- ( ( I : A --> E /\ X e. A ) -> ( I ` X ) e. { x e. ~P V | 2 <_ ( # ` x ) } )
6		fveq2 6191	. . . . 5 |- ( y = ( I ` X ) -> ( # ` y ) = ( # ` ( I ` X ) ) )
7	6	breq2d 4665	. . . 4 |- ( y = ( I ` X ) -> ( 2 <_ ( # ` y ) <-> 2 <_ ( # ` ( I ` X ) ) ) )
8		fveq2 6191	. . . . . 6 |- ( x = y -> ( # ` x ) = ( # ` y ) )
9	8	breq2d 4665	. . . . 5 |- ( x = y -> ( 2 <_ ( # ` x ) <-> 2 <_ ( # ` y ) ) )
10	9	cbvrabv 3199	. . . 4 |- { x e. ~P V | 2 <_ ( # ` x ) } = { y e. ~P V | 2 <_ ( # ` y ) }
11	7, 10	elrab2 3366	. . 3 |- ( ( I ` X ) e. { x e. ~P V | 2 <_ ( # ` x ) } <-> ( ( I ` X ) e. ~P V /\ 2 <_ ( # ` ( I ` X ) ) ) )
12	11	simprbi 480	. 2 |- ( ( I ` X ) e. { x e. ~P V | 2 <_ ( # ` x ) } -> 2 <_ ( # ` ( I ` X ) ) )
13	5, 12	syl 17	1 |- ( ( I : A --> E /\ X e. A ) -> 2 <_ ( # ` ( I ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  iEdgciedg 25875

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-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

This theorem is referenced by:  lfgrnloop  26020