Theorem upgredg 16299

Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)

Hypotheses

Ref	Expression
upgredg.v	|- V = (Vtx ` G )
upgredg.e	|- E = (Edg ` G )

Assertion

Ref	Expression
upgredg	|- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )

Distinct variable groups:   C, a, b   G, a, b   V, a, b

Allowed substitution hints:   E( a, b)

Proof of Theorem upgredg

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . 5 |- E = (Edg ` G )
2		edgvalg 16214	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2283	. . . 4 |- ( G e. UPGraph -> E = ran (iEdg ` G ) )
4	3	eleq2d 2308	. . 3 |- ( G e. UPGraph -> ( C e. E <-> C e. ran (iEdg ` G ) ) )
5	4	biimpa 296	. 2 |- ( ( G e. UPGraph /\ C e. E ) -> C e. ran (iEdg ` G ) )
6		upgredg.v	. . . . . . . 8 |- V = (Vtx ` G )
7		eqid 2238	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
8	6, 7	upgrfen 16252	. . . . . . 7 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
9	8	ffnd 5529	. . . . . 6 |- ( G e. UPGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
10	9	adantr 276	. . . . 5 |- ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
11		fnfun 5473	. . . . 5 |- ( (iEdg ` G ) Fn dom (iEdg ` G ) -> Fun (iEdg ` G ) )
12	10, 11	syl 14	. . . 4 |- ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) -> Fun (iEdg ` G ) )
13		simpr 110	. . . 4 |- ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) -> C e. ran (iEdg ` G ) )
14		elrnrexdm 5838	. . . 4 |- ( Fun (iEdg ` G ) -> ( C e. ran (iEdg ` G ) -> E. z e. dom (iEdg ` G ) C = ( (iEdg ` G ) ` z ) ) )
15	12, 13, 14	sylc 62	. . 3 |- ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) -> E. z e. dom (iEdg ` G ) C = ( (iEdg ` G ) ` z ) )
16		simpll 531	. . . . 5 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> G e. UPGraph )
17	16, 9	syl 14	. . . . 5 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
18		simprl 535	. . . . 5 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> z e. dom (iEdg ` G ) )
19	6, 7	upgrex 16258	. . . . 5 |- ( ( G e. UPGraph /\ (iEdg ` G ) Fn dom (iEdg ` G ) /\ z e. dom (iEdg ` G ) ) -> E. a e. V E. b e. V ( (iEdg ` G ) ` z ) = { a , b } )
20	16, 17, 18, 19	syl3anc 1278	. . . 4 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> E. a e. V E. b e. V ( (iEdg ` G ) ` z ) = { a , b } )
21		simprr 537	. . . . . 6 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> C = ( (iEdg ` G ) ` z ) )
22	21	eqeq1d 2247	. . . . 5 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> ( C = { a , b } <-> ( (iEdg ` G ) ` z ) = { a , b } ) )
23	22	2rexbidv 2575	. . . 4 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> ( E. a e. V E. b e. V C = { a , b } <-> E. a e. V E. b e. V ( (iEdg ` G ) ` z ) = { a , b } ) )
24	20, 23	mpbird 167	. . 3 |- ( ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) /\ ( z e. dom (iEdg ` G ) /\ C = ( (iEdg ` G ) ` z ) ) ) -> E. a e. V E. b e. V C = { a , b } )
25	15, 24	rexlimddv 2673	. 2 |- ( ( G e. UPGraph /\ C e. ran (iEdg ` G ) ) -> E. a e. V E. b e. V C = { a , b } )
26	5, 25	syldan 282	1 |- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   ~Pcpw 3685   {cpr 3706   class class class wbr 4125   dom cdm 4769   ran crn 4770   Fun wfun 5366   Fn wfn 5367   ` cfv 5372   1oc1o 6670   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UPGraphcupgr 16246

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-upgren 16248

This theorem is referenced by:  upgrpredgv  16301  upgredg2vtx  16303  upgredgpr  16304  usgr1vr  16403