Theorem usgredg4 15978

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 15977	. . 3 |- ( ( G e. USGraph /\ X e. dom E ) -> E. x e. V E. z e. V ( x =/= z /\ ( E ` X ) = { x , z } ) )
4		eleq2 2273	. . . . . . . 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 3724	. . . . . . . . . . . . 13 |- ( y = z -> { x , y } = { x , z } )
10	9	eqeq2d 2221	. . . . . . . . . . . 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 2210	. . . . . . . . . . 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 2893	. . . . . . . . . 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 3723	. . . . . . . . . . . 12 |- ( Y = x -> { Y , y } = { x , y } )
16	14, 15	eqeqan12rd 2226	. . . . . . . . . . 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 2511	. . . . . . . . . 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 3724	. . . . . . . . . . . . 13 |- ( y = x -> { z , y } = { z , x } )
23	22	eqeq2d 2221	. . . . . . . . . . . 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 3722	. . . . . . . . . . . 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 2893	. . . . . . . . . 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 3723	. . . . . . . . . . . 12 |- ( Y = z -> { Y , y } = { z , y } )
29	14, 28	eqeqan12rd 2226	. . . . . . . . . . 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 2511	. . . . . . . . . 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 720	. . . . . . 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 3669	. . . . . . 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 2636	. . 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 1205	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 712   /\ w3a 983   = wceq 1375   e. wcel 2180   =/= wne 2380   E.wrex 2489   {cpr 3647   dom cdm 4696   ` cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  USGraphcusgr 15917

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-1o 6532  df-2o 6533  df-er 6650  df-en 6858  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-edg 15824  df-umgren 15859  df-usgren 15919

This theorem is referenced by:  usgredgreu  15979