Theorem subgrprop3 16112

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 2231	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
4		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
5		subgrprop3.e	. . . 4 |- E = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16110	. . 3 |- ( S SubGraph G -> ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) /\ E C_ ~P V ) )
7		3simpa 1020	. . 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 14	. 2 |- ( S SubGraph G -> ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) )
9		simprl 531	. . 3 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> V C_ A )
10		rnss 4962	. . . . 5 |- ( (iEdg ` S ) C_ (iEdg ` G ) -> ran (iEdg ` S ) C_ ran (iEdg ` G ) )
11	10	ad2antll 491	. . . 4 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> ran (iEdg ` S ) C_ ran (iEdg ` G ) )
12		subgrv 16106	. . . . . 6 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
13		edgval 15910	. . . . . . . . 9 |- (Edg ` S ) = ran (iEdg ` S )
14	13	a1i 9	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> (Edg ` S ) = ran (iEdg ` S ) )
15	5, 14	eqtrid 2276	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> E = ran (iEdg ` S ) )
16		subgrprop3.b	. . . . . . . 8 |- B = (Edg ` G )
17		edgval 15910	. . . . . . . . 9 |- (Edg ` G ) = ran (iEdg ` G )
18	17	a1i 9	. . . . . . . 8 |- ( ( S e. _V /\ G e. _V ) -> (Edg ` G ) = ran (iEdg ` G ) )
19	16, 18	eqtrid 2276	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> B = ran (iEdg ` G ) )
20	15, 19	sseq12d 3258	. . . . . 6 |- ( ( S e. _V /\ G e. _V ) -> ( E C_ B <-> ran (iEdg ` S ) C_ ran (iEdg ` G ) ) )
21	12, 20	syl 14	. . . . 5 |- ( S SubGraph G -> ( E C_ B <-> ran (iEdg ` S ) C_ ran (iEdg ` G ) ) )
22	21	adantr 276	. . . 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 167	. . 3 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> E C_ B )
24	9, 23	jca 306	. 2 |- ( ( S SubGraph G /\ ( V C_ A /\ (iEdg ` S ) C_ (iEdg ` G ) ) ) -> ( V C_ A /\ E C_ B ) )
25	8, 24	mpdan 421	1 |- ( S SubGraph G -> ( V C_ A /\ E C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   ran crn 4726   ` 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-in1 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  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-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-edgf 15855  df-iedg 15865  df-edg 15908  df-subgr 16104

This theorem is referenced by: (None)