Theorem upgrex 16327

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 16325	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) )
4		en1 7080	. . . . . . 7 |- ( ( E ` F ) ~~ 1o <-> E. z ( E ` F ) = { z } )
5		dfsn2 3722	. . . . . . . . 9 |- { z } = { z , z }
6	5	eqeq2i 2249	. . . . . . . 8 |- ( ( E ` F ) = { z } <-> ( E ` F ) = { z , z } )
7	6	exbii 1658	. . . . . . 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 3788	. . . . . . . . . . 11 |- ( y = z -> { z , y } = { z , z } )
10	9	eqeq2d 2250	. . . . . . . . . 10 |- ( y = z -> ( ( E ` F ) = { z , y } <-> ( E ` F ) = { z , z } ) )
11	10	spcegv 2913	. . . . . . . . 9 |- ( z e. _V -> ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } ) )
12	11	elv 2825	. . . . . . . 8 |- ( ( E ` F ) = { z , z } -> E. y ( E ` F ) = { z , y } )
13		preq1 3787	. . . . . . . . . . . 12 |- ( x = z -> { x , y } = { z , y } )
14	13	eqeq2d 2250	. . . . . . . . . . 11 |- ( x = z -> ( ( E ` F ) = { x , y } <-> ( E ` F ) = { z , y } ) )
15	14	exbidv 1878	. . . . . . . . . 10 |- ( x = z -> ( E. y ( E ` F ) = { x , y } <-> E. y ( E ` F ) = { z , y } ) )
16	15	spcegv 2913	. . . . . . . . 9 |- ( z e. _V -> ( E. y ( E ` F ) = { z , y } -> E. x E. y ( E ` F ) = { x , y } ) )
17	16	elv 2825	. . . . . . . 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 1651	. . . . . 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 7106	. . . . 5 |- ( ( E ` F ) ~~ 2o -> E. x E. y ( E ` F ) = { x , y } )
22	20, 21	jaoi 728	. . . 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 1028	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> G e. UPGraph )
25		simp3 1030	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. A )
26		fndm 5478	. . . . . . . . . . 11 |- ( E Fn A -> dom E = A )
27	26	3ad2ant2 1050	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> dom E = A )
28	25, 27	eleqtrrd 2318	. . . . . . . . 9 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. dom E )
29	1, 2	upgrss 16323	. . . . . . . . 9 |- ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )
30	24, 28, 29	syl2anc 415	. . . . . . . 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 2824	. . . . . . . . 9 |- x e. _V
33	32	prid1 3816	. . . . . . . 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 2328	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. ( E ` F ) )
36	31, 35	sseldd 3249	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> x e. V )
37		vex 2824	. . . . . . . . 9 |- y e. _V
38	37	prid2 3817	. . . . . . . 8 |- y e. { x , y }
39	38, 34	eleqtrrid 2328	. . . . . . 7 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( E ` F ) = { x , y } ) -> y e. ( E ` F ) )
40	31, 39	sseldd 3249	. . . . . 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 1935	. . 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 2570	. 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 720   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   C_ wss 3220   {csn 3708   {cpr 3709   class class class wbr 4128   dom cdm 4772   Fn wfn 5370   ` cfv 5375   1oc1o 6674   2oc2o 6675   ~~ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-en 7017  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-upgren 16317

This theorem is referenced by:  upgredg  16368