Theorem issubgr 26163

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 6191	. . . . . . 7 |- ( s = S -> (Vtx ` s ) = (Vtx ` S ) )
2	1	adantr 481	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` s ) = (Vtx ` S ) )
3		fveq2 6191	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4	3	adantl 482	. . . . . 6 |- ( ( s = S /\ g = G ) -> (Vtx ` g ) = (Vtx ` G ) )
5	2, 4	sseq12d 3634	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Vtx ` s ) C_ (Vtx ` g ) <-> (Vtx ` S ) C_ (Vtx ` G ) ) )
6		fveq2 6191	. . . . . . 7 |- ( s = S -> (iEdg ` s ) = (iEdg ` S ) )
7	6	adantr 481	. . . . . 6 |- ( ( s = S /\ g = G ) -> (iEdg ` s ) = (iEdg ` S ) )
8		fveq2 6191	. . . . . . . 8 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
9	8	adantl 482	. . . . . . 7 |- ( ( s = S /\ g = G ) -> (iEdg ` g ) = (iEdg ` G ) )
10	6	dmeqd 5326	. . . . . . . 8 |- ( s = S -> dom (iEdg ` s ) = dom (iEdg ` S ) )
11	10	adantr 481	. . . . . . 7 |- ( ( s = S /\ g = G ) -> dom (iEdg ` s ) = dom (iEdg ` S ) )
12	9, 11	reseq12d 5397	. . . . . 6 |- ( ( s = S /\ g = G ) -> ( (iEdg ` g ) |` dom (iEdg ` s ) ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
13	7, 12	eqeq12d 2637	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) ) )
14		fveq2 6191	. . . . . . 7 |- ( s = S -> (Edg ` s ) = (Edg ` S ) )
15	1	pweqd 4163	. . . . . . 7 |- ( s = S -> ~P (Vtx ` s ) = ~P (Vtx ` S ) )
16	14, 15	sseq12d 3634	. . . . . 6 |- ( s = S -> ( (Edg ` s ) C_ ~P (Vtx ` s ) <-> (Edg ` S ) C_ ~P (Vtx ` S ) ) )
17	16	adantr 481	. . . . 5 |- ( ( s = S /\ g = G ) -> ( (Edg ` s ) C_ ~P (Vtx ` s ) <-> (Edg ` S ) C_ ~P (Vtx ` S ) ) )
18	5, 13, 17	3anbi123d 1399	. . . 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 26160	. . . 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 4989	. . 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 469	. 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 3631	. . 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 5325	. . . . 5 |- dom I = dom (iEdg ` S )
28	26, 27	reseq12i 5394	. . . 4 |- ( B |` dom I ) = ( (iEdg ` G ) |` dom (iEdg ` S ) )
29	25, 28	eqeq12i 2636	. . 3 |- ( I = ( B |` dom I ) <-> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
30		issubgr.e	. . . 4 |- E = (Edg ` S )
31	22	pweqi 4162	. . . 4 |- ~P V = ~P (Vtx ` S )
32	30, 31	sseq12i 3631	. . 3 |- ( E C_ ~P V <-> (Edg ` S ) C_ ~P (Vtx ` S ) )
33	24, 29, 32	3anbi123i 1251	. 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	syl6bbr 278	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   <-> 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   ` 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-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-xp 5120  df-dm 5124  df-res 5126  df-iota 5851  df-fv 5896  df-subgr 26160

This theorem is referenced by:  issubgr2  26164  subgrprop  26165  uhgrissubgr  26167  egrsubgr  26169  0grsubgr  26170  uhgrspan1  26195