Theorem issubgr2 16108

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 16107	. . 3 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
7	6	3adant2 1042	. 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 5037	. . . . 5 |- ( B |` dom I ) C_ B
9		sseq1 3250	. . . . 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 5369	. . . . . . 7 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
12	11	eqcomd 2237	. . . . . 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 1045	. . . 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 1353	. 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 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   |` cres 4727   Fun wfun 5320   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907   SubGraph csubgr 16103

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-v 2804  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-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-subgr 16104

This theorem is referenced by:  uhgrspansubgr  16127