Theorem usgredg4 16021

Description: For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)

Hypotheses

Ref	Expression
usgredg3.v	|- V = (Vtx ` G )
usgredg3.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
usgredg4	|- ( ( G e. USGraph /\ X e. dom E /\ Y e. ( E ` X ) ) -> E. y e. V ( E ` X ) = { Y , y } )

Distinct variable groups:   y, E   y, G   y, V   y, X   y, Y

Proof of Theorem usgredg4

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg3.v	. . . 4 |- V = (Vtx ` G )
2		usgredg3.e	. . . 4 |- E = (iEdg ` G )
3	1, 2	usgredg3 16020	. . 3 |- ( ( G e. USGraph /\ X e. dom E ) -> E. x e. V E. z e. V ( x =/= z /\ ( E ` X ) = { x , z } ) )
4		eleq2 2293	. . . . . . . 8 |- ( ( E ` X ) = { x , z } -> ( Y e. ( E ` X ) <-> Y e. { x , z } ) )
5	4	adantl 277	. . . . . . 7 |- ( ( x =/= z /\ ( E ` X ) = { x , z } ) -> ( Y e. ( E ` X ) <-> Y e. { x , z } ) )
6	5	adantl 277	. . . . . 6 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> ( Y e. ( E ` X ) <-> Y e. { x , z } ) )
7		simplrr 536	. . . . . . . . . . . 12 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> z e. V )
8	7	adantl 277	. . . . . . . . . . 11 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> z e. V )
9		preq2 3744	. . . . . . . . . . . . 13 |- ( y = z -> { x , y } = { x , z } )
10	9	eqeq2d 2241	. . . . . . . . . . . 12 |- ( y = z -> ( { x , z } = { x , y } <-> { x , z } = { x , z } ) )
11	10	adantl 277	. . . . . . . . . . 11 |- ( ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) /\ y = z ) -> ( { x , z } = { x , y } <-> { x , z } = { x , z } ) )
12		eqidd 2230	. . . . . . . . . . 11 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> { x , z } = { x , z } )
13	8, 11, 12	rspcedvd 2913	. . . . . . . . . 10 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> E. y e. V { x , z } = { x , y } )
14		simprr 531	. . . . . . . . . . . 12 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> ( E ` X ) = { x , z } )
15		preq1 3743	. . . . . . . . . . . 12 |- ( Y = x -> { Y , y } = { x , y } )
16	14, 15	eqeqan12rd 2246	. . . . . . . . . . 11 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> ( ( E ` X ) = { Y , y } <-> { x , z } = { x , y } ) )
17	16	rexbidv 2531	. . . . . . . . . 10 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> ( E. y e. V ( E ` X ) = { Y , y } <-> E. y e. V { x , z } = { x , y } ) )
18	13, 17	mpbird 167	. . . . . . . . 9 |- ( ( Y = x /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> E. y e. V ( E ` X ) = { Y , y } )
19	18	ex 115	. . . . . . . 8 |- ( Y = x -> ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> E. y e. V ( E ` X ) = { Y , y } ) )
20		simplrl 535	. . . . . . . . . . . 12 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> x e. V )
21	20	adantl 277	. . . . . . . . . . 11 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> x e. V )
22		preq2 3744	. . . . . . . . . . . . 13 |- ( y = x -> { z , y } = { z , x } )
23	22	eqeq2d 2241	. . . . . . . . . . . 12 |- ( y = x -> ( { x , z } = { z , y } <-> { x , z } = { z , x } ) )
24	23	adantl 277	. . . . . . . . . . 11 |- ( ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) /\ y = x ) -> ( { x , z } = { z , y } <-> { x , z } = { z , x } ) )
25		prcom 3742	. . . . . . . . . . . 12 |- { x , z } = { z , x }
26	25	a1i 9	. . . . . . . . . . 11 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> { x , z } = { z , x } )
27	21, 24, 26	rspcedvd 2913	. . . . . . . . . 10 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> E. y e. V { x , z } = { z , y } )
28		preq1 3743	. . . . . . . . . . . 12 |- ( Y = z -> { Y , y } = { z , y } )
29	14, 28	eqeqan12rd 2246	. . . . . . . . . . 11 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> ( ( E ` X ) = { Y , y } <-> { x , z } = { z , y } ) )
30	29	rexbidv 2531	. . . . . . . . . 10 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> ( E. y e. V ( E ` X ) = { Y , y } <-> E. y e. V { x , z } = { z , y } ) )
31	27, 30	mpbird 167	. . . . . . . . 9 |- ( ( Y = z /\ ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) ) -> E. y e. V ( E ` X ) = { Y , y } )
32	31	ex 115	. . . . . . . 8 |- ( Y = z -> ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> E. y e. V ( E ` X ) = { Y , y } ) )
33	19, 32	jaoi 721	. . . . . . 7 |- ( ( Y = x \/ Y = z ) -> ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> E. y e. V ( E ` X ) = { Y , y } ) )
34		elpri 3689	. . . . . . 7 |- ( Y e. { x , z } -> ( Y = x \/ Y = z ) )
35	33, 34	syl11 31	. . . . . 6 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> ( Y e. { x , z } -> E. y e. V ( E ` X ) = { Y , y } ) )
36	6, 35	sylbid 150	. . . . 5 |- ( ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) /\ ( x =/= z /\ ( E ` X ) = { x , z } ) ) -> ( Y e. ( E ` X ) -> E. y e. V ( E ` X ) = { Y , y } ) )
37	36	ex 115	. . . 4 |- ( ( ( G e. USGraph /\ X e. dom E ) /\ ( x e. V /\ z e. V ) ) -> ( ( x =/= z /\ ( E ` X ) = { x , z } ) -> ( Y e. ( E ` X ) -> E. y e. V ( E ` X ) = { Y , y } ) ) )
38	37	rexlimdvva 2656	. . 3 |- ( ( G e. USGraph /\ X e. dom E ) -> ( E. x e. V E. z e. V ( x =/= z /\ ( E ` X ) = { x , z } ) -> ( Y e. ( E ` X ) -> E. y e. V ( E ` X ) = { Y , y } ) ) )
39	3, 38	mpd 13	. 2 |- ( ( G e. USGraph /\ X e. dom E ) -> ( Y e. ( E ` X ) -> E. y e. V ( E ` X ) = { Y , y } ) )
40	39	3impia 1224	1 |- ( ( G e. USGraph /\ X e. dom E /\ Y e. ( E ` X ) ) -> E. y e. V ( E ` X ) = { Y , y } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   {cpr 3667   dom cdm 4719   ` cfv 5318  Vtxcvtx 15821  iEdgciedg 15822  USGraphcusgr 15960

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  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 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sub 8327  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-dec 9587  df-ndx 13043  df-slot 13044  df-base 13046  df-edgf 15814  df-vtx 15823  df-iedg 15824  df-edg 15867  df-umgren 15902  df-usgren 15962

This theorem is referenced by:  usgredgreu  16022