Theorem upgrop 16025

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 2231	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2231	. . 3 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgrfen 16018	. 2 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } )
4		vtxex 15939	. . . . 5 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
5		iedgex 15940	. . . . 5 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
6		opexg 4326	. . . . 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 2231	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
9		eqid 2231	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
10	8, 9	isupgren 16016	. . . 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 15946	. . . . 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 4939	. . . 4 |- ( G e. UPGraph -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15		opvtxfv 15943	. . . . . . 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 3661	. . . . 5 |- ( G e. UPGraph -> ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ~P (Vtx ` G ) )
18	17	rabeqdv 2797	. . . 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 5480	. . 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 2202   {crab 2515   _Vcvv 2803   ~Pcpw 3656   <.cop 3676   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: (None)