Theorem issubgr2 26164

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 26163	. . 3 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
7	6	3adant2 1080	. 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 5422	. . . . 5 |- ( B |` dom I ) C_ B
9		sseq1 3626	. . . . 5 |- ( I = ( B |` dom I ) -> ( I C_ B <-> ( B |` dom I ) C_ B ) )
10	8, 9	mpbiri 248	. . . 4 |- ( I = ( B |` dom I ) -> I C_ B )
11		funssres 5930	. . . . . . 7 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
12	11	eqcomd 2628	. . . . . 6 |- ( ( Fun B /\ I C_ B ) -> I = ( B |` dom I ) )
13	12	ex 450	. . . . 5 |- ( Fun B -> ( I C_ B -> I = ( B |` dom I ) ) )
14	13	3ad2ant2 1083	. . . 4 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( I C_ B -> I = ( B |` dom I ) ) )
15	10, 14	impbid2 216	. . 3 |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( I = ( B |` dom I ) <-> I C_ B ) )
16	15	3anbi2d 1404	. 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 268	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   SubGraph csubgr 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-subgr 26160

This theorem is referenced by:  uhgrspansubgr  26183