Theorem isupgren 16016

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 16014	. . 3 |- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
2	1	eleq2i 2298	. 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 5648	. . . . 5 |- ( h = G -> (iEdg ` h ) = (iEdg ` G ) )
4		isupgr.e	. . . . 5 |- E = (iEdg ` G )
5	3, 4	eqtr4di 2282	. . . 4 |- ( h = G -> (iEdg ` h ) = E )
6	3	dmeqd 4939	. . . . 5 |- ( h = G -> dom (iEdg ` h ) = dom (iEdg ` G ) )
7	4	eqcomi 2235	. . . . . 6 |- (iEdg ` G ) = E
8	7	dmeqi 4938	. . . . 5 |- dom (iEdg ` G ) = dom E
9	6, 8	eqtrdi 2280	. . . 4 |- ( h = G -> dom (iEdg ` h ) = dom E )
10		fveq2 5648	. . . . . . 7 |- ( h = G -> (Vtx ` h ) = (Vtx ` G ) )
11		isupgr.v	. . . . . . 7 |- V = (Vtx ` G )
12	10, 11	eqtr4di 2282	. . . . . 6 |- ( h = G -> (Vtx ` h ) = V )
13	12	pweqd 3661	. . . . 5 |- ( h = G -> ~P (Vtx ` h ) = ~P V )
14	13	rabeqdv 2797	. . . 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 5480	. . 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 15939	. . . . . . 7 |- ( g e. _V -> (Vtx ` g ) e. _V )
17	16	elv 2807	. . . . . 6 |- (Vtx ` g ) e. _V
18	17	a1i 9	. . . . 5 |- ( g = h -> (Vtx ` g ) e. _V )
19		fveq2 5648	. . . . 5 |- ( g = h -> (Vtx ` g ) = (Vtx ` h ) )
20		iedgex 15940	. . . . . . . 8 |- ( g e. _V -> (iEdg ` g ) e. _V )
21	20	elv 2807	. . . . . . 7 |- (iEdg ` g ) e. _V
22	21	a1i 9	. . . . . 6 |- ( ( g = h /\ v = (Vtx ` h ) ) -> (iEdg ` g ) e. _V )
23		fveq2 5648	. . . . . . 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 4939	. . . . . . 7 |- ( ( ( g = h /\ v = (Vtx ` h ) ) /\ e = (iEdg ` h ) ) -> dom e = dom (iEdg ` h ) )
27		pweq 3659	. . . . . . . . 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 2797	. . . . . . 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 5480	. . . . . 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 3070	. . . . 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 3070	. . . 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 2357	. . 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 2954	. 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 2202   {cab 2217   {crab 2515   _Vcvv 2803   [.wsbc 3032   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   -->wf 5329   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15933  iEdgciedg 15934  UPGraphcupgr 16012

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8395  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-dec 9655  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15926  df-vtx 15935  df-iedg 15936  df-upgren 16014

This theorem is referenced by:  wrdupgren  16017  upgrfen  16018  upgrop  16025  umgrupgr  16033  upgr1edc  16042  upgrun  16047  uspgrupgr  16102  subupgr  16194