Theorem upgrop 25989

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 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . 3 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgrf 25981	. 2 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } )
4		fvex 6201	. . . 4 |- (Vtx ` G ) e. _V
5		fvex 6201	. . . 4 |- (iEdg ` G ) e. _V
6	4, 5	pm3.2i 471	. . 3 |- ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V )
7		opex 4932	. . . . 5 |- <. (Vtx ` G ) , (iEdg ` G ) >. e. _V
8		eqid 2622	. . . . . 6 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
9		eqid 2622	. . . . . 6 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
10	8, 9	isupgr 25979	. . . . 5 |- ( <. (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 ) <_ 2 } ) )
11	7, 10	mp1i 13	. . . 4 |- ( ( (Vtx ` G ) e. _V /\ (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 ) <_ 2 } ) )
12		opiedgfv 25887	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
13	12	dmeqd 5326	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
14		opvtxfv 25884	. . . . . . . 8 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
15	14	pweqd 4163	. . . . . . 7 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ~P (Vtx ` G ) )
16	15	difeq1d 3727	. . . . . 6 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> ( ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) \ { (/) } ) = ( ~P (Vtx ` G ) \ { (/) } ) )
17	16	rabeqdv 3194	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> { p e. ( ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) \ { (/) } ) | ( # ` p ) <_ 2 } = { p e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } )
18	12, 13, 17	feq123d 6034	. . . 4 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) : dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) --> { p e. ( ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) \ { (/) } ) | ( # ` p ) <_ 2 } <-> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
19	11, 18	bitrd 268	. . 3 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
20	6, 19	mp1i 13	. 2 |- ( G e. UPGraph -> ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
21	3, 20	mpbird 247	1 |- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-upgr 25977

This theorem is referenced by:  finsumvtxdg2size  26446