Theorem upgrex 15924

Description: An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
upgrex	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x e. V E. y e. V ( E ` F ) = { x , y } )

Distinct variable groups:   x, G   x, V   x, E   x, F   x, A, y   y, E   y, F   y, G   y, V

Proof of Theorem upgrex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 |- V = (Vtx ` G )
2		isupgr.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	upgr1or2 15922	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )
4		en1 6964	. . . . . . 7 |- ( ( E ` F ) ~~ 1o <-> E. z ( E ` F ) = { z } )
5		dfsn2 3680	. . . . . . . . 9 |- { z } = { z , z }
6	5	eqeq2i 2240	. . . . . . . 8 |- ( ( E ` F ) = { z } <-> ( E ` F ) = { z , z } )
7	6	exbii 1651	. . . . . . 7 |- ( E. z ( E ` F ) = { z } <-> E. z ( E ` F ) = { z , z } )
8	4, 7	bitri 184	. . . . . 6 |- ( ( E ` F ) ~~ 1o <-> E. z ( E ` F ) = { z , z } )
9		preq2 3744	. . . . . . . . . . 11 |- ( y = z -> { z , y } = { z , z } )
10	9	eqeq2d 2241	. . . . . . . . . 10 |- ( y = z -> ( ( E ` F ) = { z , y } <-> ( E ` F ) = { z , z } ) )
11	10	spcegv 2891	. . . . . . . . 9 |- ( z e. _V -> ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } ) )
12	11	elv 2803	. . . . . . . 8 |- ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } )
13		preq1 3743	. . . . . . . . . . . 12 |- ( x = z -> { x , y } = { z , y } )
14	13	eqeq2d 2241	. . . . . . . . . . 11 |- ( x = z -> ( ( E ` F ) = { x , y } <-> ( E ` F ) = { z , y } ) )
15	14	exbidv 1871	. . . . . . . . . 10 |- ( x = z -> ( E. y ( E ` F ) = { x , y } <-> E. y ( E ` F ) = { z , y } ) )
16	15	spcegv 2891	. . . . . . . . 9 |- ( z e. _V -> ( E. y ( E ` F ) = { z , y } -> E. x E. y ( E ` F ) = { x , y } ) )
17	16	elv 2803	. . . . . . . 8 |- ( E. y ( E ` F ) = { z , y } -> E. x E. y ( E ` F ) = { x , y } )
18	12, 17	syl 14	. . . . . . 7 |- ( ( E ` F ) = { z , z } -> E. x E. y ( E ` F ) = { x , y } )
19	18	exlimiv 1644	. . . . . 6 |- ( E. z ( E ` F ) = { z , z } -> E. x E. y ( E ` F ) = { x , y } )
20	8, 19	sylbi 121	. . . . 5 |- ( ( E ` F ) ~~ 1o -> E. x E. y ( E ` F ) = { x , y } )
21		en2 6986	. . . . 5 |- ( ( E ` F ) ~~ 2o -> E. x E. y ( E ` F ) = { x , y } )
22	20, 21	jaoi 721	. . . 4 |- ( ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) -> E. x E. y ( E ` F ) = { x , y } )
23	3, 22	syl 14	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x E. y ( E ` F ) = { x , y } )
24		simp1 1021	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> G e. UPGraph )
25		simp3 1023	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. A )
26		fndm 5423	. . . . . . . . . . 11 |- ( E Fn A -> dom E = A )
27	26	3ad2ant2 1043	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> dom E = A )
28	25, 27	eleqtrrd 2309	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. dom E )
29	1, 2	upgrss 15920	. . . . . . . . 9 |- ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )
30	24, 28, 29	syl2anc 411	. . . . . . . 8 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) C_ V )
31	30	adantr 276	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> ( E ` F ) C_ V )
32		vex 2802	. . . . . . . . 9 |- x e. _V
33	32	prid1 3772	. . . . . . . 8 |- x e. { x , y }
34		simpr 110	. . . . . . . 8 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> ( E ` F ) = { x , y } )
35	33, 34	eleqtrrid 2319	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. ( E ` F ) )
36	31, 35	sseldd 3225	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. V )
37		vex 2802	. . . . . . . . 9 |- y e. _V
38	37	prid2 3773	. . . . . . . 8 |- y e. { x , y }
39	38, 34	eleqtrrid 2319	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> y e. ( E ` F ) )
40	31, 39	sseldd 3225	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> y e. V )
41	36, 40, 34	jca31 309	. . . . 5 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> ( ( x e. V /\ y e. V ) /\ ( E ` F ) = { x , y } ) )
42	41	ex 115	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) = { x , y } -> ( ( x e. V /\ y e. V ) /\ ( E ` F ) = { x , y } ) ) )
43	42	2eximdv 1928	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E. x E. y ( E ` F ) = { x , y } -> E. x E. y ( ( x e. V /\ y e. V ) /\ ( E ` F ) = { x , y } ) ) )
44	23, 43	mpd 13	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x E. y ( ( x e. V /\ y e. V ) /\ ( E ` F ) = { x , y } ) )
45		r2ex 2550	. 2 |- ( E. x e. V E. y e. V ( E ` F ) = { x , y } <-> E. x E. y ( ( x e. V /\ y e. V ) /\ ( E ` F ) = { x , y } ) )
46	44, 45	sylibr 134	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x e. V E. y e. V ( E ` F ) = { x , y } )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   _Vcvv 2799   C_ wss 3197   {csn 3666   {cpr 3667   class class class wbr 4083   dom cdm 4720   Fn wfn 5316   ` cfv 5321   1oc1o 6566   2oc2o 6567   ~~ cen 6898  Vtxcvtx 15834  iEdgciedg 15835  UPGraphcupgr 15912

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 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126

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 4385  df-iord 4458  df-on 4460  df-suc 4463  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-1o 6573  df-2o 6574  df-en 6901  df-sub 8335  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-dec 9595  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-upgren 15914

This theorem is referenced by:  upgredg  15963