Theorem issubgr2 16379

Description: The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-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
issubgr2	|- ( ( G e. W /\ Fun B /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B /\ E C_ ~P V ) ) )

Proof of Theorem issubgr2

Step	Hyp	Ref	Expression
1		issubgr.v	. . . 4 |- V = (Vtx ` S )
2		issubgr.a	. . . 4 |- A = (Vtx ` G )
3		issubgr.i	. . . 4 |- I = (iEdg ` S )
4		issubgr.b	. . . 4 |- B = (iEdg ` G )
5		issubgr.e	. . . 4 |- E = (Edg ` S )
6	1, 2, 3, 4, 5	issubgr 16378	. . 3 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
7	6	3adant2 1043	. 2 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
8		resss 5067	. . . . 5 |- ( B |` dom I ) C_ B
9		sseq1 3265	. . . . 5 |- ( I = ( B |` dom I ) -> ( I C_ B <-> ( B |` dom I ) C_ B ) )
10	8, 9	mpbiri 168	. . . 4 |- ( I = ( B |` dom I ) -> I C_ B )
11		funssres 5400	. . . . . . 7 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
12	11	eqcomd 2240	. . . . . 6 |- ( ( Fun B /\ I C_ B ) -> I = ( B |` dom I ) )
13	12	ex 115	. . . . 5 |- ( Fun B -> ( I C_ B -> I = ( B |` dom I ) ) )
14	13	3ad2ant2 1046	. . . 4 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( I C_ B -> I = ( B |` dom I ) ) )
15	10, 14	impbid2 143	. . 3 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( I = ( B |` dom I ) <-> I C_ B ) )
16	15	3anbi2d 1354	. 2 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) <-> ( V C_ A /\ I C_ B /\ E C_ ~P V ) ) )
17	7, 16	bitrd 188	1 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B /\ E C_ ~P V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   |` cres 4756   Fun wfun 5351   ` cfv 5357  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178   SubGraph csubgr 16374

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365  df-subgr 16375

This theorem is referenced by:  uhgrspansubgr  16398