Theorem usgredg2v 16268

Description: In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgredg2v.v	|- V = (Vtx ` G )
usgredg2v.e	|- E = (iEdg ` G )
usgredg2v.a	|- A = { x e. dom E | N e. ( E ` x ) }
usgredg2v.f	|- F = ( y e. A |-> ( iota_ z e. V ( E ` y ) = { z , N } ) )

Assertion

Ref	Expression
usgredg2v	|- ( ( G e. USGraph /\ N e. V ) -> F : A -1-1-> V )

Distinct variable groups:   x, E, z   z, G   x, N, z   z, V   y, A   y, E, x, z   y, G   y, N   y, V

Allowed substitution hints:   A( x, z)   F( x, y, z)   G( x)   V( x)

Proof of Theorem usgredg2v

Dummy variables w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg2v.v	. . . . 5 |- V = (Vtx ` G )
2		usgredg2v.e	. . . . 5 |- E = (iEdg ` G )
3		usgredg2v.a	. . . . 5 |- A = { x e. dom E | N e. ( E ` x ) }
4	1, 2, 3	usgredg2vlem1 16266	. . . 4 |- ( ( G e. USGraph /\ y e. A ) -> ( iota_ z e. V ( E ` y ) = { z , N } ) e. V )
5	4	ralrimiva 2617	. . 3 |- ( G e. USGraph -> A. y e. A ( iota_ z e. V ( E ` y ) = { z , N } ) e. V )
6	5	adantr 276	. 2 |- ( ( G e. USGraph /\ N e. V ) -> A. y e. A ( iota_ z e. V ( E ` y ) = { z , N } ) e. V )
7		simpr 110	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) )
8		preq1 3770	. . . . . . . . . 10 |- ( u = z -> { u , N } = { z , N } )
9	8	eqeq2d 2246	. . . . . . . . 9 |- ( u = z -> ( ( E ` y ) = { u , N } <-> ( E ` y ) = { z , N } ) )
10	9	cbvriotavw 6016	. . . . . . . 8 |- ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ z e. V ( E ` y ) = { z , N } )
11	8	eqeq2d 2246	. . . . . . . . 9 |- ( u = z -> ( ( E ` w ) = { u , N } <-> ( E ` w ) = { z , N } ) )
12	11	cbvriotavw 6016	. . . . . . . 8 |- ( iota_ u e. V ( E ` w ) = { u , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } )
13	7, 10, 12	3eqtr4g 2292	. . . . . . 7 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) )
14		eqid 2234	. . . . . . 7 |- N = N
15	13, 14	jctir 313	. . . . . 6 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) )
16	15	orcd 741	. . . . 5 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) )
17		simpl 109	. . . . . . . . . 10 |- ( ( G e. USGraph /\ N e. V ) -> G e. USGraph )
18		simpl 109	. . . . . . . . . 10 |- ( ( y e. A /\ w e. A ) -> y e. A )
19	17, 18	anim12i 338	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( G e. USGraph /\ y e. A ) )
20	1, 2, 3	usgredg2vlem2 16267	. . . . . . . . 9 |- ( ( G e. USGraph /\ y e. A ) -> ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ z e. V ( E ` y ) = { z , N } ) -> ( E ` y ) = { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } ) )
21	19, 10, 20	mpisyl 1492	. . . . . . . 8 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( E ` y ) = { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } )
22		an3 591	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( G e. USGraph /\ w e. A ) )
23	1, 2, 3	usgredg2vlem2 16267	. . . . . . . . 9 |- ( ( G e. USGraph /\ w e. A ) -> ( ( iota_ u e. V ( E ` w ) = { u , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) -> ( E ` w ) = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } ) )
24	22, 12, 23	mpisyl 1492	. . . . . . . 8 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( E ` w ) = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } )
25	21, 24	eqeq12d 2249	. . . . . . 7 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( ( E ` y ) = ( E ` w ) <-> { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } ) )
26	2	usgrf1 16219	. . . . . . . . 9 |- ( G e. USGraph -> E : dom E -1-1-> ran E )
27	26	adantr 276	. . . . . . . 8 |- ( ( G e. USGraph /\ N e. V ) -> E : dom E -1-1-> ran E )
28		elrabi 2972	. . . . . . . . . 10 |- ( y e. { x e. dom E | N e. ( E ` x ) } -> y e. dom E )
29	28, 3	eleq2s 2329	. . . . . . . . 9 |- ( y e. A -> y e. dom E )
30		elrabi 2972	. . . . . . . . . 10 |- ( w e. { x e. dom E | N e. ( E ` x ) } -> w e. dom E )
31	30, 3	eleq2s 2329	. . . . . . . . 9 |- ( w e. A -> w e. dom E )
32	29, 31	anim12i 338	. . . . . . . 8 |- ( ( y e. A /\ w e. A ) -> ( y e. dom E /\ w e. dom E ) )
33		f1fveq 5947	. . . . . . . 8 |- ( ( E : dom E -1-1-> ran E /\ ( y e. dom E /\ w e. dom E ) ) -> ( ( E ` y ) = ( E ` w ) <-> y = w ) )
34	27, 32, 33	syl2an 289	. . . . . . 7 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( ( E ` y ) = ( E ` w ) <-> y = w ) )
35		vtxex 16062	. . . . . . . . . . . 12 |- ( G e. USGraph -> (Vtx ` G ) e. _V )
36	1, 35	eqeltrid 2321	. . . . . . . . . . 11 |- ( G e. USGraph -> V e. _V )
37		riotaexg 6009	. . . . . . . . . . 11 |- ( V e. _V -> ( iota_ u e. V ( E ` y ) = { u , N } ) e. _V )
38	36, 37	syl 14	. . . . . . . . . 10 |- ( G e. USGraph -> ( iota_ u e. V ( E ` y ) = { u , N } ) e. _V )
39	38	adantr 276	. . . . . . . . 9 |- ( ( G e. USGraph /\ N e. V ) -> ( iota_ u e. V ( E ` y ) = { u , N } ) e. _V )
40		simpr 110	. . . . . . . . 9 |- ( ( G e. USGraph /\ N e. V ) -> N e. V )
41		riotaexg 6009	. . . . . . . . . . 11 |- ( V e. _V -> ( iota_ u e. V ( E ` w ) = { u , N } ) e. _V )
42	36, 41	syl 14	. . . . . . . . . 10 |- ( G e. USGraph -> ( iota_ u e. V ( E ` w ) = { u , N } ) e. _V )
43	42	adantr 276	. . . . . . . . 9 |- ( ( G e. USGraph /\ N e. V ) -> ( iota_ u e. V ( E ` w ) = { u , N } ) e. _V )
44		preq12bg 3879	. . . . . . . . 9 |- ( ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) e. _V /\ N e. V ) /\ ( ( iota_ u e. V ( E ` w ) = { u , N } ) e. _V /\ N e. V ) ) -> ( { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } <-> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) ) )
45	39, 40, 43, 40, 44	syl22anc 1275	. . . . . . . 8 |- ( ( G e. USGraph /\ N e. V ) -> ( { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } <-> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) ) )
46	45	adantr 276	. . . . . . 7 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( { ( iota_ u e. V ( E ` y ) = { u , N } ) , N } = { ( iota_ u e. V ( E ` w ) = { u , N } ) , N } <-> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) ) )
47	25, 34, 46	3bitr3d 218	. . . . . 6 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( y = w <-> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) ) )
48	47	adantr 276	. . . . 5 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> ( y = w <-> ( ( ( iota_ u e. V ( E ` y ) = { u , N } ) = ( iota_ u e. V ( E ` w ) = { u , N } ) /\ N = N ) \/ ( ( iota_ u e. V ( E ` y ) = { u , N } ) = N /\ N = ( iota_ u e. V ( E ` w ) = { u , N } ) ) ) ) )
49	16, 48	mpbird 167	. . . 4 |- ( ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) /\ ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) ) -> y = w )
50	49	ex 115	. . 3 |- ( ( ( G e. USGraph /\ N e. V ) /\ ( y e. A /\ w e. A ) ) -> ( ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) -> y = w ) )
51	50	ralrimivva 2626	. 2 |- ( ( G e. USGraph /\ N e. V ) -> A. y e. A A. w e. A ( ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) -> y = w ) )
52		usgredg2v.f	. . 3 |- F = ( y e. A |-> ( iota_ z e. V ( E ` y ) = { z , N } ) )
53		fveqeq2 5681	. . . 4 |- ( y = w -> ( ( E ` y ) = { z , N } <-> ( E ` w ) = { z , N } ) )
54	53	riotabidv 6007	. . 3 |- ( y = w -> ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) )
55	52, 54	f1mpt 5946	. 2 |- ( F : A -1-1-> V <-> ( A. y e. A ( iota_ z e. V ( E ` y ) = { z , N } ) e. V /\ A. y e. A A. w e. A ( ( iota_ z e. V ( E ` y ) = { z , N } ) = ( iota_ z e. V ( E ` w ) = { z , N } ) -> y = w ) ) )
56	6, 51, 55	sylanbrc 417	1 |- ( ( G e. USGraph /\ N e. V ) -> F : A -1-1-> V )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   {cpr 3692   |-> cmpt 4173   dom cdm 4751   ran crn 4752   -1-1->wf1 5351   ` cfv 5354   iota_crio 6004  Vtxcvtx 16056  iEdgciedg 16057  USGraphcusgr 16198

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-edg 16102  df-umgren 16138  df-usgren 16200

This theorem is referenced by:  usgriedgdomord  16269