Theorem upgredg 15818

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 15741	. . . . 5 |- ( G e. UPGraph -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2251	. . . 4 |- ( G e. UPGraph -> E = ran (iEdg ` G ) )
4	3	eleq2d 2276	. . 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 2206	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
8	6, 7	upgrfen 15778	. . . . . . 7 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
9	8	ffnd 5441	. . . . . 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 5385	. . . . 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 5737	. . . 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 15784	. . . . 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 1250	. . . 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 2215	. . . . 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 2532	. . . 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 2629	. 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 710   = wceq 1373   e. wcel 2177   E.wrex 2486   {crab 2489   ~Pcpw 3621   {cpr 3639   class class class wbr 4054   dom cdm 4688   ran crn 4689   Fun wfun 5279   Fn wfn 5280   ` cfv 5285   1oc1o 6513   2oc2o 6514   ~~ cen 6843  Vtxcvtx 15696  iEdgciedg 15697  Edgcedg 15739  UPGraphcupgr 15772

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 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066

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 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-1o 6520  df-2o 6521  df-en 6846  df-sub 8275  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-9 9132  df-n0 9326  df-dec 9535  df-ndx 12920  df-slot 12921  df-base 12923  df-edgf 15689  df-vtx 15698  df-iedg 15699  df-edg 15740  df-upgren 15774

This theorem is referenced by:  upgrpredgv  15820  upgredg2vtx  15822  upgredgpr  15823