Theorem uspgredg2v 16474

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 16473	. . . 4 |- ( ( G e. USPGraph /\ y e. A ) -> ( iota_ z e. V y = { N , z } ) e. V )
5	4	ralrimiva 2623	. . 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 3789	. . . . . . 7 |- ( z = n -> { N , z } = { N , n } )
8	7	eqeq2d 2250	. . . . . 6 |- ( z = n -> ( y = { N , z } <-> y = { N , n } ) )
9	8	cbvriotavw 6049	. . . . 5 |- ( iota_ z e. V y = { N , z } ) = ( iota_ n e. V y = { N , n } )
10	7	eqeq2d 2250	. . . . . 6 |- ( z = n -> ( x = { N , z } <-> x = { N , n } ) )
11	10	cbvriotavw 6049	. . . . 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 2300	. . . . . . . . . . 11 |- ( e = y -> ( N e. e <-> N e. y ) )
14	13, 3	elrab2 2985	. . . . . . . . . 10 |- ( y e. A <-> ( y e. E /\ N e. y ) )
15	2	eleq2i 2305	. . . . . . . . . . . 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 1013	. . . . . . 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 16471	. . . . . . 7 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) -> E! n e. (Vtx ` G ) y = { N , n } )
24		reueq1 2751	. . . . . . . 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 2300	. . . . . . . . . . 11 |- ( e = x -> ( N e. e <-> N e. x ) )
29	28, 3	elrab2 2985	. . . . . . . . . 10 |- ( x e. A <-> ( x e. E /\ N e. x ) )
30	2	eleq2i 2305	. . . . . . . . . . . 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 1013	. . . . . . 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 16471	. . . . . . 7 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) -> E! n e. (Vtx ` G ) x = { N , n } )
39		reueq1 2751	. . . . . . . 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 6063	. . . 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 2632	. 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 2245	. . . 4 |- ( y = x -> ( y = { N , z } <-> x = { N , z } ) )
48	47	riotabidv 6040	. . 3 |- ( y = x -> ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) )
49	46, 48	f1mpt 5977	. 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 421	1 |- ( ( G e. USPGraph /\ N e. V ) -> F : A -1-1-> V )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E!wreu 2530   {crab 2532   {cpr 3710   |-> cmpt 4192   -1-1->wf1 5374   ` cfv 5377   iota_crio 6037  Vtxcvtx 16265  Edgcedg 16310  USPGraphcuspgr 16406

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8499  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-dec 9780  df-ndx 13357  df-slot 13358  df-base 13360  df-edgf 16258  df-vtx 16267  df-iedg 16268  df-edg 16311  df-upgren 16346  df-uspgren 16408

This theorem is used by:  uspgredgdomord  16482