Theorem uspgredg2v 26116

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 26115	. . . 4 |- ( ( G e. USPGraph /\ y e. A ) -> ( iota_ z e. V y = { N , z } ) e. V )
5	4	ralrimiva 2966	. . 3 |- ( G e. USPGraph -> A. y e. A ( iota_ z e. V y = { N , z } ) e. V )
6	5	adantr 481	. 2 |- ( ( G e. USPGraph /\ N e. V ) -> A. y e. A ( iota_ z e. V y = { N , z } ) e. V )
7		preq2 4269	. . . . . . 7 |- ( z = n -> { N , z } = { N , n } )
8	7	eqeq2d 2632	. . . . . 6 |- ( z = n -> ( y = { N , z } <-> y = { N , n } ) )
9	8	cbvriotav 6622	. . . . 5 |- ( iota_ z e. V y = { N , z } ) = ( iota_ n e. V y = { N , n } )
10	7	eqeq2d 2632	. . . . . 6 |- ( z = n -> ( x = { N , z } <-> x = { N , n } ) )
11	10	cbvriotav 6622	. . . . 5 |- ( iota_ z e. V x = { N , z } ) = ( iota_ n e. V x = { N , n } )
12		simpl 473	. . . . . . . 8 |- ( ( G e. USPGraph /\ N e. V ) -> G e. USPGraph )
13		eleq2 2690	. . . . . . . . . . 11 |- ( e = y -> ( N e. e <-> N e. y ) )
14	13, 3	elrab2 3366	. . . . . . . . . 10 |- ( y e. A <-> ( y e. E /\ N e. y ) )
15	2	eleq2i 2693	. . . . . . . . . . . 12 |- ( y e. E <-> y e. (Edg ` G ) )
16	15	biimpi 206	. . . . . . . . . . 11 |- ( y e. E -> y e. (Edg ` G ) )
17	16	anim1i 592	. . . . . . . . . 10 |- ( ( y e. E /\ N e. y ) -> ( y e. (Edg ` G ) /\ N e. y ) )
18	14, 17	sylbi 207	. . . . . . . . 9 |- ( y e. A -> ( y e. (Edg ` G ) /\ N e. y ) )
19	18	adantr 481	. . . . . . . 8 |- ( ( y e. A /\ x e. A ) -> ( y e. (Edg ` G ) /\ N e. y ) )
20	12, 19	anim12i 590	. . . . . . 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 1042	. . . . . . 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 224	. . . . . 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 26112	. . . . . . 7 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) -> E! n e. (Vtx ` G ) y = { N , n } )
24		reueq1 3140	. . . . . . . 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 224	. . . . . 6 |- ( ( G e. USPGraph /\ y e. (Edg ` G ) /\ N e. y ) -> E! n e. V y = { N , n } )
27	22, 26	syl 17	. . . . 5 |- ( ( ( G e. USPGraph /\ N e. V ) /\ ( y e. A /\ x e. A ) ) -> E! n e. V y = { N , n } )
28		eleq2 2690	. . . . . . . . . . 11 |- ( e = x -> ( N e. e <-> N e. x ) )
29	28, 3	elrab2 3366	. . . . . . . . . 10 |- ( x e. A <-> ( x e. E /\ N e. x ) )
30	2	eleq2i 2693	. . . . . . . . . . . 12 |- ( x e. E <-> x e. (Edg ` G ) )
31	30	biimpi 206	. . . . . . . . . . 11 |- ( x e. E -> x e. (Edg ` G ) )
32	31	anim1i 592	. . . . . . . . . 10 |- ( ( x e. E /\ N e. x ) -> ( x e. (Edg ` G ) /\ N e. x ) )
33	29, 32	sylbi 207	. . . . . . . . 9 |- ( x e. A -> ( x e. (Edg ` G ) /\ N e. x ) )
34	33	adantl 482	. . . . . . . 8 |- ( ( y e. A /\ x e. A ) -> ( x e. (Edg ` G ) /\ N e. x ) )
35	12, 34	anim12i 590	. . . . . . 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 1042	. . . . . . 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 224	. . . . . 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 26112	. . . . . . 7 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) -> E! n e. (Vtx ` G ) x = { N , n } )
39		reueq1 3140	. . . . . . . 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 224	. . . . . 6 |- ( ( G e. USPGraph /\ x e. (Edg ` G ) /\ N e. x ) -> E! n e. V x = { N , n } )
42	37, 41	syl 17	. . . . 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 6634	. . . 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 450	. . 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 2971	. 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 2626	. . . 4 |- ( y = x -> ( y = { N , z } <-> x = { N , z } ) )
48	47	riotabidv 6613	. . 3 |- ( y = x -> ( iota_ z e. V y = { N , z } ) = ( iota_ z e. V x = { N , z } ) )
49	46, 48	f1mpt 6518	. 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 698	1 |- ( ( G e. USPGraph /\ N e. V ) -> F : A -1-1-> V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   {crab 2916   {cpr 4179   |-> cmpt 4729   -1-1->wf1 5885   ` cfv 5888   iota_crio 6610  Vtxcvtx 25874  Edgcedg 25939   USPGraph cuspgr 26043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-upgr 25977  df-uspgr 26045

This theorem is referenced by:  uspgredgleord  26124