Theorem lfgredg2dom 15982

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 5477	. . . 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 5782	. 2 |- ( ( I : A --> E /\ X e. A ) -> ( I ` X ) e. { x e. ~P V | 2o ~<_ x } )
6		breq2 4092	. . . 4 |- ( y = ( I ` X ) -> ( 2o ~<_ y <-> 2o ~<_ ( I ` X ) ) )
7		breq2 4092	. . . . 5 |- ( x = y -> ( 2o ~<_ x <-> 2o ~<_ y ) )
8	7	cbvrabv 2801	. . . 4 |- { x e. ~P V | 2o ~<_ x } = { y e. ~P V | 2o ~<_ y }
9	6, 8	elrab2 2965	. . 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 1397   e. wcel 2202   {crab 2514   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   -->wf 5322   ` cfv 5326   2oc2o 6575   ~<_ cdom 6907  iEdgciedg 15863

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  lfgrnloopen  15983