Theorem usgredg4 16069

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 16068	. . 3 |- ( ( G e. USGraph /\ X e. dom E ) -> E. x e. V E. z e. V ( x =/= z /\ ( E ` X ) = { x , z } ) )
4		eleq2 2295	. . . . . . . 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 538	. . . . . . . . . . . 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 3749	. . . . . . . . . . . . 13 |- ( y = z -> { x , y } = { x , z } )
10	9	eqeq2d 2243	. . . . . . . . . . . 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 2232	. . . . . . . . . . 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 2916	. . . . . . . . . 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 533	. . . . . . . . . . . 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 3748	. . . . . . . . . . . 12 |- ( Y = x -> { Y , y } = { x , y } )
16	14, 15	eqeqan12rd 2248	. . . . . . . . . . 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 2533	. . . . . . . . . 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 537	. . . . . . . . . . . 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 3749	. . . . . . . . . . . . 13 |- ( y = x -> { z , y } = { z , x } )
23	22	eqeq2d 2243	. . . . . . . . . . . 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 3747	. . . . . . . . . . . 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 2916	. . . . . . . . . 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 3748	. . . . . . . . . . . 12 |- ( Y = z -> { Y , y } = { z , y } )
29	14, 28	eqeqan12rd 2248	. . . . . . . . . . 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 2533	. . . . . . . . . 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 723	. . . . . . 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 3692	. . . . . . 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 2658	. . 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 1226	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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   {cpr 3670   dom cdm 4725   ` cfv 5326  Vtxcvtx 15866  iEdgciedg 15867  USGraphcusgr 16008

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-umgren 15948  df-usgren 16010

This theorem is referenced by:  usgredgreu  16070