Theorem lfgredg2dom 15924

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 2229	. . . . 5 |- A = A
2		lfuhgrnloopv.e	. . . . 5 |- E = { x e. ~P V | 2o ~<_ x }
3	1, 2	feq23i 5467	. . . 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 5769	. 2 |- ( ( I : A --> E /\ X e. A ) -> ( I ` X ) e. { x e. ~P V | 2o ~<_ x } )
6		breq2 4086	. . . 4 |- ( y = ( I ` X ) -> ( 2o ~<_ y <-> 2o ~<_ ( I ` X ) ) )
7		breq2 4086	. . . . 5 |- ( x = y -> ( 2o ~<_ x <-> 2o ~<_ y ) )
8	7	cbvrabv 2798	. . . 4 |- { x e. ~P V | 2o ~<_ x } = { y e. ~P V | 2o ~<_ y }
9	6, 8	elrab2 2962	. . 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 1395   e. wcel 2200   {crab 2512   ~Pcpw 3649   class class class wbr 4082   dom cdm 4718   -->wf 5313   ` cfv 5317   2oc2o 6554   ~<_ cdom 6884  iEdgciedg 15808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325

This theorem is referenced by:  lfgrnloopen  15925