Theorem uspgrsprf1 41755

Description: The mapping F is a one-to-one function from the "simple pseudographs" with a fixed set of vertices V into the subsets of the set of pairs over the set V. (Contributed by AV, 25-Nov-2021.)

Hypotheses

Ref	Expression
uspgrsprf.p	|- P = ~P (Pairs ` V )
uspgrsprf.g	|- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) }
uspgrsprf.f	|- F = ( g e. G |-> ( 2nd ` g ) )

Assertion

Ref	Expression
uspgrsprf1	|- F : G -1-1-> P

Distinct variable groups:   P, e, q, v   e, V, q, v   g, G   P, g, e, v

Allowed substitution hints:   F( v, e, g, q)   G( v, e, q)   V( g)

Proof of Theorem uspgrsprf1

Dummy variables a b f w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrsprf.p	. . 3 |- P = ~P (Pairs ` V )
2		uspgrsprf.g	. . 3 |- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) }
3		uspgrsprf.f	. . 3 |- F = ( g e. G |-> ( 2nd ` g ) )
4	1, 2, 3	uspgrsprf 41754	. 2 |- F : G --> P
5	1, 2, 3	uspgrsprfv 41753	. . . . 5 |- ( a e. G -> ( F ` a ) = ( 2nd ` a ) )
6	1, 2, 3	uspgrsprfv 41753	. . . . 5 |- ( b e. G -> ( F ` b ) = ( 2nd ` b ) )
7	5, 6	eqeqan12d 2638	. . . 4 |- ( ( a e. G /\ b e. G ) -> ( ( F ` a ) = ( F ` b ) <-> ( 2nd ` a ) = ( 2nd ` b ) ) )
8	2	eleq2i 2693	. . . . . 6 |- ( a e. G <-> a e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } )
9		elopab 4983	. . . . . 6 |- ( a e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } <-> E. v E. e ( a = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) )
10		opeq12 4404	. . . . . . . . 9 |- ( ( v = w /\ e = f ) -> <. v , e >. = <. w , f >. )
11	10	eqeq2d 2632	. . . . . . . 8 |- ( ( v = w /\ e = f ) -> ( a = <. v , e >. <-> a = <. w , f >. ) )
12		eqeq1 2626	. . . . . . . . . 10 |- ( v = w -> ( v = V <-> w = V ) )
13	12	adantr 481	. . . . . . . . 9 |- ( ( v = w /\ e = f ) -> ( v = V <-> w = V ) )
14		eqeq2 2633	. . . . . . . . . . 11 |- ( v = w -> ( (Vtx ` q ) = v <-> (Vtx ` q ) = w ) )
15		eqeq2 2633	. . . . . . . . . . 11 |- ( e = f -> ( (Edg ` q ) = e <-> (Edg ` q ) = f ) )
16	14, 15	bi2anan9 917	. . . . . . . . . 10 |- ( ( v = w /\ e = f ) -> ( ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) <-> ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) )
17	16	rexbidv 3052	. . . . . . . . 9 |- ( ( v = w /\ e = f ) -> ( E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) <-> E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) )
18	13, 17	anbi12d 747	. . . . . . . 8 |- ( ( v = w /\ e = f ) -> ( ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) <-> ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) )
19	11, 18	anbi12d 747	. . . . . . 7 |- ( ( v = w /\ e = f ) -> ( ( a = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) <-> ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) )
20	19	cbvex2v 2287	. . . . . 6 |- ( E. v E. e ( a = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) <-> E. w E. f ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) )
21	8, 9, 20	3bitri 286	. . . . 5 |- ( a e. G <-> E. w E. f ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) )
22	2	eleq2i 2693	. . . . . 6 |- ( b e. G <-> b e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } )
23		elopab 4983	. . . . . 6 |- ( b e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } <-> E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) )
24	22, 23	bitri 264	. . . . 5 |- ( b e. G <-> E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) )
25		eqeq2 2633	. . . . . . . . . . . . . . . 16 |- ( w = V -> ( v = w <-> v = V ) )
26		opeq12 4404	. . . . . . . . . . . . . . . . . 18 |- ( ( w = v /\ f = e ) -> <. w , f >. = <. v , e >. )
27	26	ex 450	. . . . . . . . . . . . . . . . 17 |- ( w = v -> ( f = e -> <. w , f >. = <. v , e >. ) )
28	27	equcoms 1947	. . . . . . . . . . . . . . . 16 |- ( v = w -> ( f = e -> <. w , f >. = <. v , e >. ) )
29	25, 28	syl6bir 244	. . . . . . . . . . . . . . 15 |- ( w = V -> ( v = V -> ( f = e -> <. w , f >. = <. v , e >. ) ) )
30	29	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( v = V -> ( f = e -> <. w , f >. = <. v , e >. ) ) )
31	30	com12 32	. . . . . . . . . . . . 13 |- ( v = V -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( f = e -> <. w , f >. = <. v , e >. ) ) )
32	31	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( f = e -> <. w , f >. = <. v , e >. ) ) )
33	32	imp 445	. . . . . . . . . . 11 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( f = e -> <. w , f >. = <. v , e >. ) )
34		vex 3203	. . . . . . . . . . . . . 14 |- w e. _V
35		vex 3203	. . . . . . . . . . . . . 14 |- f e. _V
36	34, 35	op2ndd 7179	. . . . . . . . . . . . 13 |- ( a = <. w , f >. -> ( 2nd ` a ) = f )
37	36	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( 2nd ` a ) = f )
38		vex 3203	. . . . . . . . . . . . . . 15 |- v e. _V
39		vex 3203	. . . . . . . . . . . . . . 15 |- e e. _V
40	38, 39	op2ndd 7179	. . . . . . . . . . . . . 14 |- ( b = <. v , e >. -> ( 2nd ` b ) = e )
41	40	adantr 481	. . . . . . . . . . . . 13 |- ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( 2nd ` b ) = e )
42	41	adantr 481	. . . . . . . . . . . 12 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( 2nd ` b ) = e )
43	37, 42	eqeq12d 2637	. . . . . . . . . . 11 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) <-> f = e ) )
44		eqeq12 2635	. . . . . . . . . . . . . . . 16 |- ( ( a = <. w , f >. /\ b = <. v , e >. ) -> ( a = b <-> <. w , f >. = <. v , e >. ) )
45	44	ex 450	. . . . . . . . . . . . . . 15 |- ( a = <. w , f >. -> ( b = <. v , e >. -> ( a = b <-> <. w , f >. = <. v , e >. ) ) )
46	45	adantr 481	. . . . . . . . . . . . . 14 |- ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( b = <. v , e >. -> ( a = b <-> <. w , f >. = <. v , e >. ) ) )
47	46	com12 32	. . . . . . . . . . . . 13 |- ( b = <. v , e >. -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( a = b <-> <. w , f >. = <. v , e >. ) ) )
48	47	adantr 481	. . . . . . . . . . . 12 |- ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( a = b <-> <. w , f >. = <. v , e >. ) ) )
49	48	imp 445	. . . . . . . . . . 11 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( a = b <-> <. w , f >. = <. v , e >. ) )
50	33, 43, 49	3imtr4d 283	. . . . . . . . . 10 |- ( ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) /\ ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) )
51	50	ex 450	. . . . . . . . 9 |- ( ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) ) )
52	51	exlimivv 1860	. . . . . . . 8 |- ( E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) ) )
53	52	com12 32	. . . . . . 7 |- ( ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) ) )
54	53	exlimivv 1860	. . . . . 6 |- ( E. w E. f ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) -> ( E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) ) )
55	54	imp 445	. . . . 5 |- ( ( E. w E. f ( a = <. w , f >. /\ ( w = V /\ E. q e. USPGraph ( (Vtx ` q ) = w /\ (Edg ` q ) = f ) ) ) /\ E. v E. e ( b = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) )
56	21, 24, 55	syl2anb 496	. . . 4 |- ( ( a e. G /\ b e. G ) -> ( ( 2nd ` a ) = ( 2nd ` b ) -> a = b ) )
57	7, 56	sylbid 230	. . 3 |- ( ( a e. G /\ b e. G ) -> ( ( F ` a ) = ( F ` b ) -> a = b ) )
58	57	rgen2a 2977	. 2 |- A. a e. G A. b e. G ( ( F ` a ) = ( F ` b ) -> a = b )
59		dff13 6512	. 2 |- ( F : G -1-1-> P <-> ( F : G --> P /\ A. a e. G A. b e. G ( ( F ` a ) = ( F ` b ) -> a = b ) ) )
60	4, 58, 59	mpbir2an 955	1 |- F : G -1-1-> P

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   ~Pcpw 4158   <.cop 4183   {copab 4712   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   2ndc2nd 7167  Vtxcvtx 25874  Edgcedg 25939   USPGraph cuspgr 26043  Pairscspr 41727

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  df-spr 41728

This theorem is referenced by:  uspgrsprf1o  41757