Theorem upgredg 16014

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 15929	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2276	. . . 4 |- ( G e. UPGraph -> E = ran (iEdg ` G ) )
4	3	eleq2d 2301	. . 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 2231	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
8	6, 7	upgrfen 15967	. . . . . . 7 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
9	8	ffnd 5483	. . . . . 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 5427	. . . . 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 5786	. . . 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 531	. . . . 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 15973	. . . . 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 1273	. . . 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 533	. . . . . 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 2240	. . . . 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 2557	. . . 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 2655	. 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 715   = wceq 1397   e. wcel 2202   E.wrex 2511   {crab 2514   ~Pcpw 3652   {cpr 3670   class class class wbr 4088   dom cdm 4725   ran crn 4726   Fun wfun 5320   Fn wfn 5321   ` cfv 5326   1oc1o 6575   2oc2o 6576   ~~ cen 6907  Vtxcvtx 15882  iEdgciedg 15883  Edgcedg 15927  UPGraphcupgr 15961

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13103  df-slot 13104  df-base 13106  df-edgf 15875  df-vtx 15884  df-iedg 15885  df-edg 15928  df-upgren 15963

This theorem is referenced by:  upgrpredgv  16016  upgredg2vtx  16018  upgredgpr  16019  usgr1vr  16118