Theorem uspgredg2v 15984

Description: In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)

Hypotheses

Ref	Expression
uspgredg2v.v	|- V = (Vtx ` G )
uspgredg2v.e	|- E = (Edg ` G )
uspgredg2v.a	|- A = { e e. E | N e. e }
uspgredg2v.f	|- F = ( y e. A |-> ( iota_ z e. V y = { N , z } ) )

Assertion

Ref	Expression
uspgredg2v	|- ( ( G e. USPGraph /\ N e. V ) -> F : A -1-1-> V )

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

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

Proof of Theorem uspgredg2v

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

Step	Hyp	Ref	Expression
1		uspgredg2v.v	. . . . 5 |- V = (Vtx ` G )
2		uspgredg2v.e	. . . . 5 |- E = (Edg ` G )
3		uspgredg2v.a	. . . . 5 |- A = { e e. E | N e. e }
4	1, 2, 3	uspgredg2vlem 15983	. . . 4 |- ( ( G e. USPGraph /\ y e. A ) -> ( iota_ z e. V y = { N , z } ) e. V )
5	4	ralrimiva 2583	. . 3 |- ( G e. USPGraph -> A. y e. A ( iota_ z e. V y = { N , z } ) e. V )
6	5	adantr 276	. 2 |- ( ( G e. USPGraph /\ N e. V ) -> A. y e. A ( iota_ z e. V y = { N , z } ) e. V )
7		preq2 3724	. . . . . . 7 |- ( z = n -> { N , z } = { N , n } )
8	7	eqeq2d 2221	. . . . . 6 |- ( z = n -> ( y = { N , z } <-> y = { N , n } ) )
9	8	cbvriotavw 5938	. . . . 5 |- ( iota_ z e. V y = { N , z } ) = ( iota_ n e. V y = { N , n } )
10	7	eqeq2d 2221	. . . . . 6 |- ( z = n -> ( x = { N , z } <-> x = { N , n } ) )
11	10	cbvriotavw 5938	. . . . 5 |- ( iota_ z e. V x = { N , z } ) = ( iota_ n e. V x = { N , n } )
12		simpl 109	. . . . . . . 8 |- ( ( G e. USPGraph /\ N e. V ) -> G e. USPGraph )
13		eleq2w 2271	. . . . . . . . . . 11 |- ( e = y -> ( N e. e <-> N e. y ) )
14	13, 3	elrab2 2942	. . . . . . . . . 10 |- ( y e. A <-> ( y e. E /\ N e. y ) )
15	2	eleq2i 2276	. . . . . . . . . . . 12 |- ( y e. E <-> y e. (Edg ` G ) )
16	15	biimpi 120	. . . . . . . . . . 11 |- ( y e. E -> y e. (Edg ` G ) )
17	16	anim1i 340	. . . . . . . . . 10 |- ( ( y e. E /\ N e. y ) -> ( y e. (Edg ` G ) /\ N e. y ) )
18	14, 17	sylbi 121	. . . . . . . . 9 |- ( y e. A -> ( y e. (Edg ` G ) /\ N e. y ) )
19	18	adantr 276	. . . . . . . 8 |- ( ( y e. A /\ x e. A ) -> ( y e. (Edg ` G ) /\ N e. y ) )
20	12, 19	anim12i 338	. . . . . . 7 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> ( G e. USPGraph /\ ( y e. (Edg ` G ) /\ N e. y ) ) )
21		3anass 987	. . . . . . 7 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) <-> ( G e. USPGraph /\ ( y e. (Edg ` G ) /\ N e. y ) ) )
22	20, 21	sylibr 134	. . . . . 6 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) )
23		uspgredg2vtxeu 15981	. . . . . . 7 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) -> E! n e. (Vtx ` G ) y = { N , n } )
24		reueq1 2710	. . . . . . . 8 |- ( V = (Vtx ` G ) -> ( E! n e. V y = { N , n } <-> E! n e. (Vtx ` G ) y = { N , n } ) )
25	1, 24	ax-mp 5	. . . . . . 7 |- ( E! n e. V y = { N , n } <-> E! n e. (Vtx ` G ) y = { N , n } )
26	23, 25	sylibr 134	. . . . . 6 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) -> E! n e. V y = { N , n } )
27	22, 26	syl 14	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> E! n e. V y = { N , n } )
28		eleq2w 2271	. . . . . . . . . . 11 |- ( e = x -> ( N e. e <-> N e. x ) )
29	28, 3	elrab2 2942	. . . . . . . . . 10 |- ( x e. A <-> ( x e. E /\ N e. x ) )
30	2	eleq2i 2276	. . . . . . . . . . . 12 |- ( x e. E <-> x e. (Edg ` G ) )
31	30	biimpi 120	. . . . . . . . . . 11 |- ( x e. E -> x e. (Edg ` G ) )
32	31	anim1i 340	. . . . . . . . . 10 |- ( ( x e. E /\ N e. x ) -> ( x e. (Edg ` G ) /\ N e. x ) )
33	29, 32	sylbi 121	. . . . . . . . 9 |- ( x e. A -> ( x e. (Edg ` G ) /\ N e. x ) )
34	33	adantl 277	. . . . . . . 8 |- ( ( y e. A /\ x e. A ) -> ( x e. (Edg ` G ) /\ N e. x ) )
35	12, 34	anim12i 338	. . . . . . 7 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> ( G e. USPGraph /\ ( x e. (Edg ` G ) /\ N e. x ) ) )
36		3anass 987	. . . . . . 7 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) <-> ( G e. USPGraph /\ ( x e. (Edg ` G ) /\ N e. x ) ) )
37	35, 36	sylibr 134	. . . . . 6 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) )
38		uspgredg2vtxeu 15981	. . . . . . 7 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) -> E! n e. (Vtx ` G ) x = { N , n } )
39		reueq1 2710	. . . . . . . 8 |- ( V = (Vtx ` G ) -> ( E! n e. V x = { N , n } <-> E! n e. (Vtx ` G ) x = { N , n } ) )
40	1, 39	ax-mp 5	. . . . . . 7 |- ( E! n e. V x = { N , n } <-> E! n e. (Vtx ` G ) x = { N , n } )
41	38, 40	sylibr 134	. . . . . 6 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) -> E! n e. V x = { N , n } )
42	37, 41	syl 14	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> E! n e. V x = { N , n } )
43	9, 11, 27, 42	riotaeqimp 5952	. . . 4 |- ( ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) /\ ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) ) -> y = x )
44	43	ex 115	. . 3 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> ( ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) -> y = x ) )
45	44	ralrimivva 2592	. 2 |- ( ( G e. USPGraph /\ N e. V ) -> A. y e. A A. x e. A ( ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) -> y = x ) )
46		uspgredg2v.f	. . 3 |- F = ( y e. A |-> ( iota_ z e. V y = { N , z } ) )
47		eqeq1 2216	. . . 4 |- ( y = x -> ( y = { N , z } <-> x = { N , z } ) )
48	47	riotabidv 5929	. . 3 |- ( y = x -> ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) )
49	46, 48	f1mpt 5868	. 2 |- ( F : A -1-1-> V <-> ( A. y e. A ( iota_ z e. V y = { N , z } ) e. V /\ A. y e. A A. x e. A ( ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) -> y = x ) ) )
50	6, 45, 49	sylanbrc 417	1 |- ( ( G e. USPGraph /\ N e. V ) -> F : A -1-1-> V )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 983   = wceq 1375   e. wcel 2180   A.wral 2488   E!wreu 2490   {crab 2492   {cpr 3647   |-> cmpt 4124   -1-1->wf1 5291   ` cfv 5294   iota_crio 5926  Vtxcvtx 15778  Edgcedg 15823  USPGraphcuspgr 15916

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-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-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-rmo 2496  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-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-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-upgren 15858  df-uspgren 15918

This theorem is referenced by:  uspgredgdomord  15992