Theorem upgrex 16024

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 16022	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )
4		en1 7016	. . . . . . 7 |- ( ( E ` F ) ~~ 1o <-> E. z ( E ` F ) = { z } )
5		dfsn2 3687	. . . . . . . . 9 |- { z } = { z , z }
6	5	eqeq2i 2242	. . . . . . . 8 |- ( ( E ` F ) = { z } <-> ( E ` F ) = { z , z } )
7	6	exbii 1654	. . . . . . 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 3753	. . . . . . . . . . 11 |- ( y = z -> { z , y } = { z , z } )
10	9	eqeq2d 2243	. . . . . . . . . 10 |- ( y = z -> ( ( E ` F ) = { z , y } <-> ( E ` F ) = { z , z } ) )
11	10	spcegv 2895	. . . . . . . . 9 |- ( z e. _V -> ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } ) )
12	11	elv 2807	. . . . . . . 8 |- ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } )
13		preq1 3752	. . . . . . . . . . . 12 |- ( x = z -> { x , y } = { z , y } )
14	13	eqeq2d 2243	. . . . . . . . . . 11 |- ( x = z -> ( ( E ` F ) = { x , y } <-> ( E ` F ) = { z , y } ) )
15	14	exbidv 1873	. . . . . . . . . 10 |- ( x = z -> ( E. y ( E ` F ) = { x , y } <-> E. y ( E ` F ) = { z , y } ) )
16	15	spcegv 2895	. . . . . . . . 9 |- ( z e. _V -> ( E. y ( E ` F ) = { z , y } -> E. x E. y ( E ` F ) = { x , y } ) )
17	16	elv 2807	. . . . . . . 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 1647	. . . . . 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 7041	. . . . 5 |- ( ( E ` F ) ~~ 2o -> E. x E. y ( E ` F ) = { x , y } )
22	20, 21	jaoi 724	. . . 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 1024	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> G e. UPGraph )
25		simp3 1026	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. A )
26		fndm 5436	. . . . . . . . . . 11 |- ( E Fn A -> dom E = A )
27	26	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> dom E = A )
28	25, 27	eleqtrrd 2311	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. dom E )
29	1, 2	upgrss 16020	. . . . . . . . 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 2806	. . . . . . . . 9 |- x e. _V
33	32	prid1 3781	. . . . . . . 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 2321	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. ( E ` F ) )
36	31, 35	sseldd 3229	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. V )
37		vex 2806	. . . . . . . . 9 |- y e. _V
38	37	prid2 3782	. . . . . . . 8 |- y e. { x , y }
39	38, 34	eleqtrrid 2321	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> y e. ( E ` F ) )
40	31, 39	sseldd 3229	. . . . . 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 1930	. . 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 2553	. 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 716   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   E.wrex 2512   _Vcvv 2803   C_ wss 3201   {csn 3673   {cpr 3674   class class class wbr 4093   dom cdm 4731   Fn wfn 5328   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15933  iEdgciedg 15934  UPGraphcupgr 16012

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-en 6953  df-sub 8395  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-dec 9655  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15926  df-vtx 15935  df-iedg 15936  df-upgren 16014

This theorem is referenced by:  upgredg  16065