Theorem isusgren 16012

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 16010	. . 3 |- USGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | x ~~ 2o } }
2	1	eleq2i 2298	. 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 5639	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isuspgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2282	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4933	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2235	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4932	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2280	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5639	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isuspgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2282	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3657	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2796	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
15	5, 9, 14	f1eq123d 5575	. . 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 15872	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2806	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5639	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15873	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2806	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5639	. . . . . . 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 4933	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3655	. . . . . . . . 9 |- ( v = (Vtx ` h ) -> ~P v = ~P (Vtx ` h ) )
28	27	ad2antlr 489	. . . . . . . 8 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ~P v = ~P (Vtx ` h ) )
29	28	rabeqdv 2796	. . . . . . 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 5575	. . . . . 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 3069	. . . . 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 3069	. . . 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 2356	. . 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 2953	. 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 } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   [.wsbc 3031   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   -1-1->wf1 5323   ` cfv 5326   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15866  iEdgciedg 15867  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-usgren 16010

This theorem is referenced by:  usgrfen  16014  isusgropen  16019  ausgrusgrben  16022  ausgrusgrien  16025  usgruspgr  16037  usgrumgruspgr  16039  usgruspgrben  16040  usgrislfuspgrdom  16044  usgrstrrepeen  16085  usgr0e  16086  usgr0  16093  subusgr  16129