Theorem ausgrusgrien 16153

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 16000	. . . . 5 |- ( H e. W -> (Vtx ` H ) e. _V )
2		edgvalg 16041	. . . . . 6 |- ( H e. W -> (Edg ` H ) = ran (iEdg ` H ) )
3		iedgex 16001	. . . . . . 7 |- ( H e. W -> (iEdg ` H ) e. _V )
4		rnexg 5021	. . . . . . 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 2309	. . . . 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 16149	. . . . 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 3266	. . . . 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 2299	. . . . . . . . . 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 4955	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> dom f = dom (iEdg ` H ) )
16		rneq 4983	. . . . . . . . . . 11 |- ( f = (iEdg ` H ) -> ran f = ran (iEdg ` H ) )
17	14, 15, 16	f1eq123d 5605	. . . . . . . . . 10 |- ( f = (iEdg ` H ) -> ( f : dom f -1-1-> ran f <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> ran (iEdg ` H ) ) )
18	17	elabg 2962	. . . . . . . . 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 1047	. . . . . . 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 1025	. . . . . . 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 5579	. . . . . . 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 1229	. . . . 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 1220	. 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 2232	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
29		eqid 2232	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
30	28, 29	isusgren 16140	. . 3 |- ( H e. W -> ( H e. USGraph <-> (iEdg ` H ) : dom (iEdg ` H ) -1-1-> { x e. ~P (Vtx ` H ) | x ~~ 2o } ) )
31	30	3ad2ant1 1045	. 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 1005   = wceq 1398   e. wcel 2203   {cab 2218   {crab 2524   _Vcvv 2812   C_ wss 3210   ~Pcpw 3668   class class class wbr 4108   {copab 4169   dom cdm 4748   ran crn 4749   -1-1->wf1 5348   ` cfv 5351   2oc2o 6640   ~~ cen 6972  Vtxcvtx 15994  iEdgciedg 15995  Edgcedg 16039  USGraphcusgr 16136

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 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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-usgren 16138

This theorem is referenced by:  usgrausgrben  16154