Theorem lfgredg2dom 15808

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 2206	. . . . 5 |- A = A
2		lfuhgrnloopv.e	. . . . 5 |- E = { x e. ~P V | 2o ~<_ x }
3	1, 2	feq23i 5435	. . . 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 5733	. 2 |- ( ( I : A --> E /\ X e. A ) -> ( I ` X ) e. { x e. ~P V | 2o ~<_ x } )
6		breq2 4058	. . . 4 |- ( y = ( I ` X ) -> ( 2o ~<_ y <-> 2o ~<_ ( I ` X ) ) )
7		breq2 4058	. . . . 5 |- ( x = y -> ( 2o ~<_ x <-> 2o ~<_ y ) )
8	7	cbvrabv 2772	. . . 4 |- { x e. ~P V | 2o ~<_ x } = { y e. ~P V | 2o ~<_ y }
9	6, 8	elrab2 2936	. . 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 1373   e. wcel 2177   {crab 2489   ~Pcpw 3621   class class class wbr 4054   dom cdm 4688   -->wf 5281   ` cfv 5285   2oc2o 6514   ~<_ cdom 6844  iEdgciedg 15697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293

This theorem is referenced by:  lfgrnloopen  15809