Theorem issubgr2 16413

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 16412	. . 3 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
7	6	3adant2 1047	. 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 5082	. . . . 5 |- ( B |` dom I ) C_ B
9		sseq1 3271	. . . . 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 5415	. . . . . . 7 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
12	11	eqcomd 2244	. . . . . 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 1050	. . . 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 1358	. 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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   |` cres 4771   Fun wfun 5366   ` 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-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-subgr 16409

This theorem is referenced by:  uhgrspansubgr  16432