Theorem upgrop 16211

Description: A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)

Assertion

Ref	Expression
upgrop	|- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )

Proof of Theorem upgrop

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2234	. . 3 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgrfen 16204	. 2 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } )
4		vtxex 16125	. . . . 5 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
5		iedgex 16126	. . . . 5 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
6		opexg 4349	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> <. (Vtx ` G ) , (iEdg ` G ) >. e. _V )
7	4, 5, 6	syl2anc 411	. . . 4 |- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. _V )
8		eqid 2234	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
9		eqid 2234	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
10	8, 9	isupgren 16202	. . . 4 |- ( <. (Vtx ` G ) , (iEdg ` G ) >. e. _V -> ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph <-> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) : dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) --> { p e. ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( p ~~ 1o \/ p ~~ 2o ) } ) )
11	7, 10	syl 14	. . 3 |- ( G e. UPGraph -> ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph <-> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) : dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) --> { p e. ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( p ~~ 1o \/ p ~~ 2o ) } ) )
12		opiedgfv 16132	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
13	4, 5, 12	syl2anc 411	. . . 4 |- ( G e. UPGraph -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
14	13	dmeqd 4963	. . . 4 |- ( G e. UPGraph -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15		opvtxfv 16129	. . . . . . 7 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
16	4, 5, 15	syl2anc 411	. . . . . 6 |- ( G e. UPGraph -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
17	16	pweqd 3679	. . . . 5 |- ( G e. UPGraph -> ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ~P (Vtx ` G ) )
18	17	rabeqdv 2809	. . . 4 |- ( G e. UPGraph -> { p e. ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( p ~~ 1o \/ p ~~ 2o ) } = { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } )
19	13, 14, 18	feq123d 5504	. . 3 |- ( G e. UPGraph -> ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) : dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) --> { p e. ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( p ~~ 1o \/ p ~~ 2o ) } <-> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } ) )
20	11, 19	bitrd 188	. 2 |- ( G e. UPGraph -> ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } ) )
21	3, 20	mpbird 167	1 |- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   ~Pcpw 3674   <.cop 3697   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: (None)