Theorem subgrprop 16414

Description: The properties of a subgraph. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
issubgr.v	|- V = (Vtx ` S )
issubgr.a	|- A = (Vtx ` G )
issubgr.i	|- I = (iEdg ` S )
issubgr.b	|- B = (iEdg ` G )
issubgr.e	|- E = (Edg ` S )

Assertion

Ref	Expression
subgrprop	|- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )

Proof of Theorem subgrprop

Step	Hyp	Ref	Expression
1		subgrv 16411	. 2 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
2		issubgr.v	. . . . 5 |- V = (Vtx ` S )
3		issubgr.a	. . . . 5 |- A = (Vtx ` G )
4		issubgr.i	. . . . 5 |- I = (iEdg ` S )
5		issubgr.b	. . . . 5 |- B = (iEdg ` G )
6		issubgr.e	. . . . 5 |- E = (Edg ` S )
7	2, 3, 4, 5, 6	issubgr 16412	. . . 4 |- ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
8	7	biimpd 144	. . 3 |- ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
9	8	ancoms 268	. 2 |- ( ( S e. _V /\ G e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
10	1, 9	mpcom 36	1 |- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   |` cres 4771   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212   SubGraph csubgr 16408

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779  df-res 4781  df-iota 5332  df-fv 5380  df-subgr 16409

This theorem is referenced by:  subgrprop2  16415