Theorem ausgrusgrien 16025

Description: The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)

Hypotheses

Ref	Expression
ausgr.1	|- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
ausgrusgri.1	|- O = { f | f : dom f -1-1-> ran f }

Assertion

Ref	Expression
ausgrusgrien	|- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> H e. USGraph )

Distinct variable groups:   v, e, x, H   f, H   x, W

Allowed substitution hints:   G( x, v, e, f)   O( x, v, e, f)   W( v, e, f)

Proof of Theorem ausgrusgrien

Step	Hyp	Ref	Expression
1		vtxex 15872	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 15913	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 15873	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 4997	. . . . . . 7 |- ( (iEdg ` H ) e. _V -> ran (iEdg ` H ) e. _V )
5	3, 4	syl 14	. . . . . 6 |- ( H e. W -> ran (iEdg ` H ) e. _V )
6	2, 5	eqeltrd 2308	. . . . 5 |- ( H e. W -> (Edg ` H ) e. _V )
7		ausgr.1	. . . . . 6 |- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
8	7	isausgren 16021	. . . . 5 |- ( ( (Vtx ` H ) e. _V /\ (Edg ` H ) e. _V ) -> ( (Vtx ` H ) G (Edg ` H ) <-> (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
9	1, 6, 8	syl2anc 411	. . . 4 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) <-> (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
10	2	sseq1d 3256	. . . . 5 |- ( H e. W -> ( (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } <-> ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
11		ausgrusgri.1	. . . . . . . . . . 11 |- O = { f | f : dom f -1-1-> ran f }
12	11	eleq2i 2298	. . . . . . . . . 10 |- ( (iEdg ` H ) e. O <-> (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } )
13	12	biimpi 120	. . . . . . . . 9 |- ( (iEdg ` H ) e. O -> (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } )
14		id 19	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> f = (iEdg ` H ) )
15		dmeq 4931	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> dom f = dom (iEdg ` H ) )
16		rneq 4959	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> ran f = ran (iEdg ` H ) )
17	14, 15, 16	f1eq123d 5575	. . . . . . . . . 10 |- ( f = (iEdg ` H ) -> ( f : dom f -1-1-> ran f <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
18	17	elabg 2952	. . . . . . . . 9 |- ( (iEdg ` H ) e. O -> ( (iEdg ` H ) e. { f | f : dom f -1-1-> ran f } <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
19	13, 18	mpbid 147	. . . . . . . 8 |- ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) )
20	19	3ad2ant3 1046	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) )
21		simp2 1024	. . . . . . 7 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } )
22		f1ssr 5549	. . . . . . 7 |- ( ( (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
23	20, 21, 22	syl2anc 411	. . . . . 6 |- ( ( H e. W /\ ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
24	23	3exp 1228	. . . . 5 |- ( H e. W -> ( ran (iEdg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
25	10, 24	sylbid 150	. . . 4 |- ( H e. W -> ( (Edg ` H ) C_ { x e. ~P (Vtx ` H ) | x ~~ 2o } -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
26	9, 25	sylbid 150	. . 3 |- ( H e. W -> ( (Vtx ` H ) G (Edg ` H ) -> ( (iEdg ` H ) e. O -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) ) )
27	26	3imp 1219	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } )
28		eqid 2231	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
29		eqid 2231	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
30	28, 29	isusgren 16012	. . 3 |- ( H e. W -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
31	30	3ad2ant1 1044	. 2 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
32	27, 31	mpbird 167	1 |- ( ( H e. W /\ (Vtx ` H ) G (Edg ` H ) /\ (iEdg ` H ) e. O ) -> H e. USGraph )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   {copab 4149   dom cdm 4725   ran crn 4726   -1-1->wf1 5323   ` cfv 5326   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15866  iEdgciedg 15867  Edgcedg 15911  USGraphcusgr 16008

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

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-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-usgren 16010

This theorem is referenced by:  usgrausgrben  16026