Theorem issubgr 16107

Description: The property of a set to be a subgraph of another set. (Contributed by AV, 16-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
issubgr	|- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )

Proof of Theorem issubgr

Dummy variables s g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5639	. . . . . . 7 |- ( s = S -> (Vtx ` s ) = (Vtx ` S ) )
2	1	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` s ) = (Vtx ` S ) )
3		fveq2 5639	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4	3	adantl 277	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` g ) = (Vtx ` G ) )
5	2, 4	sseq12d 3258	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Vtx ` s ) C_ (Vtx ` g ) <-> (Vtx ` S ) C_ (Vtx ` G ) ) )
6		fveq2 5639	. . . . . . 7 |- ( s = S -> (iEdg ` s ) = (iEdg ` S ) )
7	6	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (iEdg ` s ) = (iEdg ` S ) )
8		fveq2 5639	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
9	8	adantl 277	. . . . . . 7 |- ( ( s = S /\ g = G ) -> (iEdg ` g ) = (iEdg ` G ) )
10	6	dmeqd 4933	. . . . . . . 8 |- ( s = S -> dom (iEdg ` s ) = dom (iEdg ` S ) )
11	10	adantr 276	. . . . . . 7 |- ( ( s = S /\ g = G ) -> dom (iEdg ` s ) = dom (iEdg ` S ) )
12	9, 11	reseq12d 5014	. . . . . 6 |- ( ( s = S /\ g = G ) -> ( (iEdg ` g ) |` dom (iEdg ` s ) ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
13	7, 12	eqeq12d 2246	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) ) )
14		fveq2 5639	. . . . . . 7 |- ( s = S -> (Edg ` s ) = (Edg ` S ) )
15	1	pweqd 3657	. . . . . . 7 |- ( s = S -> ~P (Vtx ` s ) = ~P (Vtx ` S ) )
16	14, 15	sseq12d 3258	. . . . . 6 |- ( s = S -> ( (Edg ` s ) C_ ~P (Vtx ` s ) <-> (Edg ` S ) C_ ~P (Vtx ` S ) ) )
17	16	adantr 276	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Edg ` s ) C_ ~P (Vtx ` s ) <-> (Edg ` S ) C_ ~P (Vtx ` S ) ) )
18	5, 13, 17	3anbi123d 1348	. . . 4 |- ( ( s = S /\ g = G ) -> ( ( (Vtx ` s ) C_ (Vtx ` g ) /\ (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) /\ (Edg ` s ) C_ ~P (Vtx ` s ) ) <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
19		df-subgr 16104	. . . 4 |- SubGraph = { <. s , g >. | ( (Vtx ` s ) C_ (Vtx ` g ) /\ (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) /\ (Edg ` s ) C_ ~P (Vtx ` s ) ) }
20	18, 19	brabga 4358	. . 3 |- ( ( S e. U /\ G e. W ) -> ( S SubGraph G <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
21	20	ancoms 268	. 2 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
22		issubgr.v	. . . 4 |- V = (Vtx ` S )
23		issubgr.a	. . . 4 |- A = (Vtx ` G )
24	22, 23	sseq12i 3255	. . 3 |- ( V C_ A <-> (Vtx ` S ) C_ (Vtx ` G ) )
25		issubgr.i	. . . 4 |- I = (iEdg ` S )
26		issubgr.b	. . . . 5 |- B = (iEdg ` G )
27	25	dmeqi 4932	. . . . 5 |- dom I = dom (iEdg ` S )
28	26, 27	reseq12i 5011	. . . 4 |- ( B |` dom I ) = ( (iEdg ` G ) |` dom (iEdg ` S ) )
29	25, 28	eqeq12i 2245	. . 3 |- ( I = ( B |` dom I ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
30		issubgr.e	. . . 4 |- E = (Edg ` S )
31	22	pweqi 3656	. . . 4 |- ~P V = ~P (Vtx ` S )
32	30, 31	sseq12i 3255	. . 3 |- ( E C_ ~P V <-> (Edg ` S ) C_ ~P (Vtx ` S ) )
33	24, 29, 32	3anbi123i 1214	. 2 |- ( ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
34	21, 33	bitr4di 198	1 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ 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   ` 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-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-xp 4731  df-dm 4735  df-res 4737  df-iota 5286  df-fv 5334  df-subgr 16104

This theorem is referenced by:  issubgr2  16108  subgrprop  16109  uhgrissubgr  16111  egrsubgr  16113  0grsubgr  16114