Theorem upgrop 15750

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 2206	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2206	. . 3 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	upgrfen 15743	. 2 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { p e. ~P (Vtx ` G ) | ( p ~~ 1o \/ p ~~ 2o ) } )
4		vtxex 15667	. . . . 5 |- ( G e. UPGraph -> (Vtx ` G ) e. _V )
5		iedgex 15668	. . . . 5 |- ( G e. UPGraph -> (iEdg ` G ) e. _V )
6		opexg 4277	. . . . 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 2206	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
9		eqid 2206	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
10	8, 9	isupgren 15741	. . . 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 15674	. . . . 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 4886	. . . 4 |- ( G e. UPGraph -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15		opvtxfv 15671	. . . . . . 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 3623	. . . . 5 |- ( G e. UPGraph -> ~P (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ~P (Vtx ` G ) )
18	17	rabeqdv 2767	. . . 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 5423	. . 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 710   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773   ~Pcpw 3618   <.cop 3638   class class class wbr 4048   dom cdm 4680   -->wf 5273   ` cfv 5277   1oc1o 6505   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UPGraphcupgr 15737

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fo 5283  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-upgren 15739

This theorem is referenced by: (None)