Theorem usgredg4 16095

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 16094	. . 3 |- ( ( G e. USGraph /\ X e. dom E ) -> E. x e. V E. z e. V ( x =/= z /\ ( E ` X ) = { x , z } ) )
4		eleq2 2294	. . . . . . . 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 3750	. . . . . . . . . . . . 13 |- ( y = z -> { x , y } = { x , z } )
10	9	eqeq2d 2242	. . . . . . . . . . . 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 2231	. . . . . . . . . . 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 2915	. . . . . . . . . 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 3749	. . . . . . . . . . . 12 |- ( Y = x -> { Y , y } = { x , y } )
16	14, 15	eqeqan12rd 2247	. . . . . . . . . . 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 2532	. . . . . . . . . 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 3750	. . . . . . . . . . . . 13 |- ( y = x -> { z , y } = { z , x } )
23	22	eqeq2d 2242	. . . . . . . . . . . 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 3748	. . . . . . . . . . . 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 2915	. . . . . . . . . 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 3749	. . . . . . . . . . . 12 |- ( Y = z -> { Y , y } = { z , y } )
29	14, 28	eqeqan12rd 2247	. . . . . . . . . . 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 2532	. . . . . . . . . 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 3693	. . . . . . 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 2657	. . 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 2201   =/= wne 2401   E.wrex 2510   {cpr 3671   dom cdm 4727   ` cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  USGraphcusgr 16034

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-1o 6587  df-2o 6588  df-er 6707  df-en 6915  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-edg 15938  df-umgren 15974  df-usgren 16036

This theorem is referenced by:  usgredgreu  16096