Theorem isupgren 16202

Description: The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, G   x, V

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

Proof of Theorem isupgren

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

Step	Hyp	Ref	Expression
1		df-upgren 16200	. . 3 |- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
2	1	eleq2i 2301	. 2 |- ( G e. UPGraph <-> G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } } )
3		fveq2 5675	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isupgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2285	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4963	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2238	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4962	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2283	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5675	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isupgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2285	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3679	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2809	. . . 4 |- ( h = G -> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
15	5, 9, 14	feq123d 5504	. . 3 |- ( h = G -> ( (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
16		vtxex 16125	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2819	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5675	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 16126	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2819	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5675	. . . . . . 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 4963	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3677	. . . . . . . . 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 2809	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } )
30	25, 26, 29	feq123d 5504	. . . . . 6 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> ( e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
31	22, 24, 30	sbcied2 3083	. . . . 5 |- ( ( g = h /\ v = (Vtx ` h ) ) -> ( [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
32	18, 19, 31	sbcied2 3083	. . . 4 |- ( g = h -> ( [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } <-> (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } ) )
33	32	cbvabv 2361	. . 3 |- { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } } = { h | (iEdg ` h ) : dom (iEdg ` h ) --> { x e. ~P (Vtx ` h ) | ( x ~~ 1o \/ x ~~ 2o ) } }
34	15, 33	elab2g 2967	. 2 |- ( G e. U -> ( G e. { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } } <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
35	2, 34	bitrid 192	1 |- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   _Vcvv 2815   [.wsbc 3045   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   -->wf 5353   ` cfv 5357   1oc1o 6653   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16119  iEdgciedg 16120  UPGraphcupgr 16198

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16112  df-vtx 16121  df-iedg 16122  df-upgren 16200

This theorem is referenced by:  wrdupgren  16203  upgrfen  16204  upgrop  16211  umgrupgr  16219  upgr1edc  16228  upgrun  16233  uspgrupgr  16288  subupgr  16380