Theorem isumgren 15751

Description: The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, G   x, V

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

Proof of Theorem isumgren

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

Step	Hyp	Ref	Expression
1		df-umgren 15740	. . 3 |- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } }
2	1	eleq2i 2273	. 2 |- ( G e. UMGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } )
3		fveq2 5586	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isumgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2257	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4886	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2210	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4885	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2255	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5586	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isumgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2257	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3623	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2767	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
15	5, 9, 14	feq123d 5423	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )
16		vtxex 15667	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2777	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5586	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15668	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2777	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5586	. . . . . . 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 4886	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3621	. . . . . . . . 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 2767	. . . . . . 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	feq123d 5423	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
31	22, 24, 30	sbcied2 3038	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
32	18, 19, 31	sbcied2 3038	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } ) )
33	32	cbvabv 2331	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | x ~~ 2o } }
34	15, 33	elab2g 2922	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } } <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )
35	2, 34	bitrid 192	1 |- ( G e. U -> ( G e. UMGraph <-> E : dom E --> { x e. ~P V | x ~~ 2o } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   {cab 2192   {crab 2489   _Vcvv 2773   [.wsbc 3000   ~Pcpw 3618   class class class wbr 4048   dom cdm 4680   -->wf 5273   ` cfv 5277   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UMGraphcumgr 15738

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fo 5283  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-umgren 15740

This theorem is referenced by:  wrdumgren  15752  umgrfen  15753  umgr0e  15761  umgrun  15769