Theorem issubgr 16269

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 5672	. . . . . . 7 |- ( s = S -> (Vtx ` s ) = (Vtx ` S ) )
2	1	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` s ) = (Vtx ` S ) )
3		fveq2 5672	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4	3	adantl 277	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` g ) = (Vtx ` G ) )
5	2, 4	sseq12d 3271	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Vtx ` s ) C_ (Vtx ` g ) <-> (Vtx ` S ) C_ (Vtx ` G ) ) )
6		fveq2 5672	. . . . . . 7 |- ( s = S -> (iEdg ` s ) = (iEdg ` S ) )
7	6	adantr 276	. . . . . 6 |- ( ( s = S /\ g = G ) -> (iEdg ` s ) = (iEdg ` S ) )
8		fveq2 5672	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
9	8	adantl 277	. . . . . . 7 |- ( ( s = S /\ g = G ) -> (iEdg ` g ) = (iEdg ` G ) )
10	6	dmeqd 4960	. . . . . . . 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 5041	. . . . . 6 |- ( ( s = S /\ g = G ) -> ( (iEdg ` g ) |` dom (iEdg ` s ) ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
13	7, 12	eqeq12d 2249	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) ) )
14		fveq2 5672	. . . . . . 7 |- ( s = S -> (Edg ` s ) = (Edg ` S ) )
15	1	pweqd 3676	. . . . . . 7 |- ( s = S -> ~P (Vtx ` s ) = ~P (Vtx ` S ) )
16	14, 15	sseq12d 3271	. . . . . 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 16266	. . . 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 4384	. . 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 3268	. . 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 4959	. . . . 5 |- dom I = dom (iEdg ` S )
28	26, 27	reseq12i 5038	. . . 4 |- ( B |` dom I ) = ( (iEdg ` G ) |` dom (iEdg ` S ) )
29	25, 28	eqeq12i 2248	. . 3 |- ( I = ( B |` dom I ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
30		issubgr.e	. . . 4 |- E = (Edg ` S )
31	22	pweqi 3675	. . . 4 |- ~P V = ~P (Vtx ` S )
32	30, 31	sseq12i 3268	. . 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 2205   C_ wss 3213   ~Pcpw 3671   class class class wbr 4111   dom cdm 4751   |` cres 4753   ` cfv 5354  Vtxcvtx 16024  iEdgciedg 16025  Edgcedg 16069   SubGraph csubgr 16265

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 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-dm 4761  df-res 4763  df-iota 5314  df-fv 5362  df-subgr 16266

This theorem is referenced by:  issubgr2  16270  subgrprop  16271  uhgrissubgr  16273  egrsubgr  16275  0grsubgr  16276