Theorem lfgredg2dom 16056

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 | 2o ~<_ x }

Assertion

Ref	Expression
lfgredg2dom	|- ( ( I : A --> E /\ X e. A ) -> 2o ~<_ ( I ` X ) )

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

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

Proof of Theorem lfgredg2dom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . 5 |- A = A
2		lfuhgrnloopv.e	. . . . 5 |- E = { x e. ~P V | 2o ~<_ x }
3	1, 2	feq23i 5484	. . . 4 |- ( I : A --> E <-> I : A --> { x e. ~P V | 2o ~<_ x } )
4	3	biimpi 120	. . 3 |- ( I : A --> E -> I : A --> { x e. ~P V | 2o ~<_ x } )
5	4	ffvelcdmda 5790	. 2 |- ( ( I : A --> E /\ X e. A ) -> ( I ` X ) e. { x e. ~P V | 2o ~<_ x } )
6		breq2 4097	. . . 4 |- ( y = ( I ` X ) -> ( 2o ~<_ y <-> 2o ~<_ ( I ` X ) ) )
7		breq2 4097	. . . . 5 |- ( x = y -> ( 2o ~<_ x <-> 2o ~<_ y ) )
8	7	cbvrabv 2802	. . . 4 |- { x e. ~P V | 2o ~<_ x } = { y e. ~P V | 2o ~<_ y }
9	6, 8	elrab2 2966	. . 3 |- ( ( I ` X ) e. { x e. ~P V | 2o ~<_ x } <-> ( ( I ` X ) e. ~P V /\ 2o ~<_ ( I ` X ) ) )
10	9	simprbi 275	. 2 |- ( ( I ` X ) e. { x e. ~P V | 2o ~<_ x } -> 2o ~<_ ( I ` X ) )
11	5, 10	syl 14	1 |- ( ( I : A --> E /\ X e. A ) -> 2o ~<_ ( I ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {crab 2515   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   -->wf 5329   ` cfv 5333   2oc2o 6619   ~<_ cdom 6951  iEdgciedg 15937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by:  lfgrnloopen  16057