Theorem isusgren 16604

Description: The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Hypotheses

Ref	Expression
isuspgr.v	|- V = (Vtx ` G )
isuspgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
isusgren	|- ( G e. U -> ( G e. USGraph <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )

Distinct variable groups:   x, G   x, V

Allowed substitution hints:   U( x)   E( x)

Proof of Theorem isusgren

Dummy variables e g h v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-usgren 16602	. . 3 |- USGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } }
2	1	eleq2i 2305	. 2 |- ( G e. USGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } } )
3		fveq2 5698	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuspgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2289	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4985	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2242	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4984	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2287	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5698	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuspgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2289	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3693	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2815	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
15	5, 9, 14	f1eq123d 5634	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { x e. ~P (Vtx ` h ) | x ~~ 2o } <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )
16		vtxex 16464	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2825	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5698	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 16465	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2825	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5698	. . . . . . 7 |- ( g = h -> (iEdg ` g ) = (iEdg ` h ) )
24	23	adantr 276	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) = (iEdg ` h ) )
25		simpr 110	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> e = (iEdg ` h ) )
26	25	dmeqd 4985	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3691	. . . . . . . . 9 |- ( v = (Vtx ` h ) -> ~P v = ~P (Vtx ` h ) )
28	27	ad2antlr 493	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2815	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P (Vtx ` h ) | x ~~ 2o } )
30	25, 26, 29	f1eq123d 5634	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e -1-1-> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
31	22, 24, 30	sbcied2 3089	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
32	18, 19, 31	sbcied2 3089	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
33	32	cbvabv 2365	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } } = { h | (iEdg ` h ) : dom (iEdg ` h ) -1-1-> { x e. ~P (Vtx ` h ) | x ~~ 2o } }
34	15, 33	elab2g 2973	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } } <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )
35	2, 34	bitrid 192	1 |- ( G e. U -> ( G e. USGraph <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   _Vcvv 2821   [.wsbc 3051   ~Pcpw 3688   class class class wbr 4131   dom cdm 4776   -1-1->wf1 5377   ` cfv 5380   2oc2o 6685   ~~ cen 7024  Vtxcvtx 16458  iEdgciedg 16459  USGraphcusgr 16600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sub 8506  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-dec 9789  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-usgren 16602

This theorem is used by:  usgrfen  16606  isusgropen  16611  ausgrusgrben  16614  ausgrusgrien  16617  usgruspgr  16629  usgrumgruspgr  16631  usgruspgrben  16632  usgrislfuspgrdom  16636  usgrstrrepeen  16677  usgr0e  16678  usgr0  16685  subusgr  16721