Theorem subgrprop3 26168

Description: The properties of a subgraph: If S is a subgraph of G, its vertices are also vertices of G, and its edges are also edges of G. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
subgrprop3.v	|- V = (Vtx ` S )
subgrprop3.a	|- A = (Vtx ` G )
subgrprop3.e	|- E = (Edg ` S )
subgrprop3.b	|- B = (Edg ` G )

Assertion

Ref	Expression
subgrprop3	|- ( S SubGraph G -> ( V C_ A /\ E C_ B ) )

Proof of Theorem subgrprop3

Step	Hyp	Ref	Expression
1		subgrprop3.v	. . . 4 |- V = (Vtx ` S )
2		subgrprop3.a	. . . 4 |- A = (Vtx ` G )
3		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		subgrprop3.e	. . . 4 |- E = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 26166	. . 3 |- ( S SubGraph G -> ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) /\ E C_ ~P V ) )
7		3simpa 1058	. . 3 |- ( ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) /\ E C_ ~P V ) -> ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
8	6, 7	syl 17	. 2 |- ( S SubGraph G -> ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
9		simprl 794	. . 3 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> V C_ A )
10		rnss 5354	. . . . 5 |- ( (iEdg ` S ) C_ (iEdg ` G ) -> ran (iEdg ` S ) C_ ran (iEdg ` G ) )
11	10	ad2antll 765	. . . 4 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> ran (iEdg ` S ) C_ ran (iEdg ` G ) )
12		subgrv 26162	. . . . . 6 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
13		edgval 25941	. . . . . . . . 9 |- (Edg ` S ) = ran (iEdg ` S )
14	13	a1i 11	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> (Edg ` S ) = ran (iEdg ` S ) )
15	5, 14	syl5eq 2668	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> E = ran (iEdg ` S ) )
16		subgrprop3.b	. . . . . . . 8 |- B = (Edg ` G )
17		edgval 25941	. . . . . . . . 9 |- (Edg ` G ) = ran (iEdg ` G )
18	17	a1i 11	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> (Edg ` G ) = ran (iEdg ` G ) )
19	16, 18	syl5eq 2668	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> B = ran (iEdg ` G ) )
20	15, 19	sseq12d 3634	. . . . . 6 |- ( ( S e. _V /\ G e. _V ) -> ( E C_ B <-> ran (iEdg ` S ) C_ ran (iEdg ` G ) ) )
21	12, 20	syl 17	. . . . 5 |- ( S SubGraph G -> ( E C_ B <-> ran (iEdg ` S ) C_ ran (iEdg ` G ) ) )
22	21	adantr 481	. . . 4 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> ( E C_ B <-> ran (iEdg ` S ) C_ ran (iEdg ` G ) ) )
23	11, 22	mpbird 247	. . 3 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> E C_ B )
24	9, 23	jca 554	. 2 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> ( V C_ A /\ E C_ B ) )
25	8, 24	mpdan 702	1 |- ( S SubGraph G -> ( V C_ A /\ E C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ran crn 5115   ` 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-edg 25940  df-subgr 26160

This theorem is referenced by: (None)