Theorem issubgr 16181

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 5648	. . . . . . 7 |- ( s = S -> (Vtx ` s ) = (Vtx ` S ) )
2	1	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` s ) = (Vtx ` S ) )
3		fveq2 5648	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4	3	adantl 277	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` g ) = (Vtx ` G ) )
5	2, 4	sseq12d 3259	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Vtx ` s ) C_ (Vtx ` g ) <-> (Vtx ` S ) C_ (Vtx ` G ) ) )
6		fveq2 5648	. . . . . . 7 |- ( s = S -> (iEdg ` s ) = (iEdg ` S ) )
7	6	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (iEdg ` s ) = (iEdg ` S ) )
8		fveq2 5648	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
9	8	adantl 277	. . . . . . 7 |- ( ( s = S /\ g = G ) -> (iEdg ` g ) = (iEdg ` G ) )
10	6	dmeqd 4939	. . . . . . . 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 5020	. . . . . 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 5648	. . . . . . 7 |- ( s = S -> (Edg ` s ) = (Edg ` S ) )
15	1	pweqd 3661	. . . . . . 7 |- ( s = S -> ~P (Vtx ` s ) = ~P (Vtx ` S ) )
16	14, 15	sseq12d 3259	. . . . . 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 1349	. . . 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 16178	. . . 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 4364	. . 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 3256	. . 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 4938	. . . . 5 |- dom I = dom (iEdg ` S )
28	26, 27	reseq12i 5017	. . . 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 3660	. . . 4 |- ~P V = ~P (Vtx ` S )
32	30, 31	sseq12i 3256	. . 3 |- ( E C_ ~P V <-> (Edg ` S ) C_ ~P (Vtx ` S ) )
33	24, 29, 32	3anbi123i 1215	. 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 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   |` cres 4733   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981   SubGraph csubgr 16177

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741  df-res 4743  df-iota 5293  df-fv 5341  df-subgr 16178

This theorem is referenced by:  issubgr2  16182  subgrprop  16183  uhgrissubgr  16185  egrsubgr  16187  0grsubgr  16188