Theorem upgredg 15957

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 15875	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2274	. . . 4 |- ( G e. UPGraph -> E = ran (iEdg ` G ) )
4	3	eleq2d 2299	. . 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 2229	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
8	6, 7	upgrfen 15912	. . . . . . 7 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
9	8	ffnd 5474	. . . . . 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 5418	. . . . 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 5776	. . . 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 527	. . . . 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 529	. . . . 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 15918	. . . . 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 1271	. . . 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 531	. . . . . 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 2238	. . . . 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 2555	. . . 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 2653	. 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 713   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   ~Pcpw 3649   {cpr 3667   class class class wbr 4083   dom cdm 4719   ran crn 4720   Fun wfun 5312   Fn wfn 5313   ` cfv 5318   1oc1o 6561   2oc2o 6562   ~~ cen 6893  Vtxcvtx 15828  iEdgciedg 15829  Edgcedg 15873  UPGraphcupgr 15906

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-1o 6568  df-2o 6569  df-en 6896  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-edg 15874  df-upgren 15908

This theorem is referenced by:  upgrpredgv  15959  upgredg2vtx  15961  upgredgpr  15962