Theorem uspgrsprfo 41756

Description: The mapping F is a function from the "simple pseudographs" with a fixed set of vertices V onto 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
uspgrsprfo	|- ( V e. W -> F : G -onto-> P )

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

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

Proof of Theorem uspgrsprfo

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

Step	Hyp	Ref	Expression
1		uspgrsprf.p	. . . 4 |- P = ~P (Pairs ` V )
2		uspgrsprf.g	. . . 4 |- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) }
3		uspgrsprf.f	. . . 4 |- F = ( g e. G |-> ( 2nd ` g ) )
4	1, 2, 3	uspgrsprf 41754	. . 3 |- F : G --> P
5	4	a1i 11	. 2 |- ( V e. W -> F : G --> P )
6	1	eleq2i 2693	. . . . . . 7 |- ( a e. P <-> a e. ~P (Pairs ` V ) )
7		selpw 4165	. . . . . . 7 |- ( a e. ~P (Pairs ` V ) <-> a C_ (Pairs ` V ) )
8	6, 7	bitri 264	. . . . . 6 |- ( a e. P <-> a C_ (Pairs ` V ) )
9		eqidd 2623	. . . . . . . . . 10 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> V = V )
10		vex 3203	. . . . . . . . . . . . . . 15 |- a e. _V
11	10	a1i 11	. . . . . . . . . . . . . 14 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> a e. _V )
12		f1oi 6174	. . . . . . . . . . . . . . . . 17 |- ( _I |` a ) : a -1-1-onto-> a
13	12	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( _I |` a ) : a -1-1-onto-> a )
14		dmresi 5457	. . . . . . . . . . . . . . . . 17 |- dom ( _I |` a ) = a
15		f1oeq2 6128	. . . . . . . . . . . . . . . . 17 |- ( dom ( _I |` a ) = a -> ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a <-> ( _I |` a ) : a -1-1-onto-> a ) )
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a <-> ( _I |` a ) : a -1-1-onto-> a )
17	13, 16	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a )
18		sprvalpwle2 41739	. . . . . . . . . . . . . . . . 17 |- ( V e. W -> (Pairs ` V ) = { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } )
19	18	sseq2d 3633	. . . . . . . . . . . . . . . 16 |- ( V e. W -> ( a C_ (Pairs ` V ) <-> a C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
20	19	biimpac 503	. . . . . . . . . . . . . . 15 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> a C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } )
21	17, 20	jca 554	. . . . . . . . . . . . . 14 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a /\ a C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
22		f1oeq3 6129	. . . . . . . . . . . . . . 15 |- ( f = a -> ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> f <-> ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a ) )
23		sseq1 3626	. . . . . . . . . . . . . . 15 |- ( f = a -> ( f C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } <-> a C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
24	22, 23	anbi12d 747	. . . . . . . . . . . . . 14 |- ( f = a -> ( ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> f /\ f C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) <-> ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> a /\ a C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) ) )
25	11, 21, 24	elabd 3352	. . . . . . . . . . . . 13 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> E. f ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> f /\ f C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
26		resiexg 7102	. . . . . . . . . . . . . . 15 |- ( a e. _V -> ( _I |` a ) e. _V )
27	10, 26	ax-mp 5	. . . . . . . . . . . . . 14 |- ( _I |` a ) e. _V
28	27	f11o 7128	. . . . . . . . . . . . 13 |- ( ( _I |` a ) : dom ( _I |` a ) -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } <-> E. f ( ( _I |` a ) : dom ( _I |` a ) -1-1-onto-> f /\ f C_ { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
29	25, 28	sylibr 224	. . . . . . . . . . . 12 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( _I |` a ) : dom ( _I |` a ) -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } )
30	10	a1i 11	. . . . . . . . . . . . . . . 16 |- ( a C_ (Pairs ` V ) -> a e. _V )
31	30	resiexd 6480	. . . . . . . . . . . . . . 15 |- ( a C_ (Pairs ` V ) -> ( _I |` a ) e. _V )
32	31	anim2i 593	. . . . . . . . . . . . . 14 |- ( ( V e. W /\ a C_ (Pairs ` V ) ) -> ( V e. W /\ ( _I |` a ) e. _V ) )
33	32	ancoms 469	. . . . . . . . . . . . 13 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( V e. W /\ ( _I |` a ) e. _V ) )
34		isuspgrop 26056	. . . . . . . . . . . . 13 |- ( ( V e. W /\ ( _I |` a ) e. _V ) -> ( <. V , ( _I |` a ) >. e. USPGraph <-> ( _I |` a ) : dom ( _I |` a ) -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
35	33, 34	syl 17	. . . . . . . . . . . 12 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( <. V , ( _I |` a ) >. e. USPGraph <-> ( _I |` a ) : dom ( _I |` a ) -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
36	29, 35	mpbird 247	. . . . . . . . . . 11 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> <. V , ( _I |` a ) >. e. USPGraph )
37		fveq2 6191	. . . . . . . . . . . . . 14 |- ( q = <. V , ( _I |` a ) >. -> (Vtx ` q ) = (Vtx ` <. V , ( _I |` a ) >. ) )
38	37	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( q = <. V , ( _I |` a ) >. -> ( (Vtx ` q ) = V <-> (Vtx ` <. V , ( _I |` a ) >. ) = V ) )
39		fveq2 6191	. . . . . . . . . . . . . 14 |- ( q = <. V , ( _I |` a ) >. -> (Edg ` q ) = (Edg ` <. V , ( _I |` a ) >. ) )
40	39	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( q = <. V , ( _I |` a ) >. -> ( (Edg ` q ) = a <-> (Edg ` <. V , ( _I |` a ) >. ) = a ) )
41	38, 40	anbi12d 747	. . . . . . . . . . . 12 |- ( q = <. V , ( _I |` a ) >. -> ( ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) <-> ( (Vtx ` <. V , ( _I |` a ) >. ) = V /\ (Edg ` <. V , ( _I |` a ) >. ) = a ) ) )
42	41	adantl 482	. . . . . . . . . . 11 |- ( ( ( a C_ (Pairs ` V ) /\ V e. W ) /\ q = <. V , ( _I |` a ) >. ) -> ( ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) <-> ( (Vtx ` <. V , ( _I |` a ) >. ) = V /\ (Edg ` <. V , ( _I |` a ) >. ) = a ) ) )
43		opvtxfv 25884	. . . . . . . . . . . . . 14 |- ( ( V e. W /\ ( _I |` a ) e. _V ) -> (Vtx ` <. V , ( _I |` a ) >. ) = V )
44	32, 43	syl 17	. . . . . . . . . . . . 13 |- ( ( V e. W /\ a C_ (Pairs ` V ) ) -> (Vtx ` <. V , ( _I |` a ) >. ) = V )
45		edgopval 25944	. . . . . . . . . . . . . . 15 |- ( ( V e. W /\ ( _I |` a ) e. _V ) -> (Edg ` <. V , ( _I |` a ) >. ) = ran ( _I |` a ) )
46	32, 45	syl 17	. . . . . . . . . . . . . 14 |- ( ( V e. W /\ a C_ (Pairs ` V ) ) -> (Edg ` <. V , ( _I |` a ) >. ) = ran ( _I |` a ) )
47		rnresi 5479	. . . . . . . . . . . . . 14 |- ran ( _I |` a ) = a
48	46, 47	syl6eq 2672	. . . . . . . . . . . . 13 |- ( ( V e. W /\ a C_ (Pairs ` V ) ) -> (Edg ` <. V , ( _I |` a ) >. ) = a )
49	44, 48	jca 554	. . . . . . . . . . . 12 |- ( ( V e. W /\ a C_ (Pairs ` V ) ) -> ( (Vtx ` <. V , ( _I |` a ) >. ) = V /\ (Edg ` <. V , ( _I |` a ) >. ) = a ) )
50	49	ancoms 469	. . . . . . . . . . 11 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( (Vtx ` <. V , ( _I |` a ) >. ) = V /\ (Edg ` <. V , ( _I |` a ) >. ) = a ) )
51	36, 42, 50	rspcedvd 3317	. . . . . . . . . 10 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) )
52	9, 51	jca 554	. . . . . . . . 9 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( V = V /\ E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) )
53	2	eleq2i 2693	. . . . . . . . . 10 |- ( <. V , a >. e. G <-> <. V , a >. e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } )
54	30	anim1i 592	. . . . . . . . . . . 12 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( a e. _V /\ V e. W ) )
55	54	ancomd 467	. . . . . . . . . . 11 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( V e. W /\ a e. _V ) )
56		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( v = V -> ( v = V <-> V = V ) )
57	56	adantr 481	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = a ) -> ( v = V <-> V = V ) )
58		eqeq2 2633	. . . . . . . . . . . . . . 15 |- ( v = V -> ( (Vtx ` q ) = v <-> (Vtx ` q ) = V ) )
59		eqeq2 2633	. . . . . . . . . . . . . . 15 |- ( e = a -> ( (Edg ` q ) = e <-> (Edg ` q ) = a ) )
60	58, 59	bi2anan9 917	. . . . . . . . . . . . . 14 |- ( ( v = V /\ e = a ) -> ( ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) <-> ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) )
61	60	rexbidv 3052	. . . . . . . . . . . . 13 |- ( ( v = V /\ e = a ) -> ( E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) <-> E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) )
62	57, 61	anbi12d 747	. . . . . . . . . . . 12 |- ( ( v = V /\ e = a ) -> ( ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) <-> ( V = V /\ E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) ) )
63	62	opelopabga 4988	. . . . . . . . . . 11 |- ( ( V e. W /\ a e. _V ) -> ( <. V , a >. e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } <-> ( V = V /\ E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) ) )
64	55, 63	syl 17	. . . . . . . . . 10 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( <. V , a >. e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } <-> ( V = V /\ E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) ) )
65	53, 64	syl5bb 272	. . . . . . . . 9 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( <. V , a >. e. G <-> ( V = V /\ E. q e. USPGraph ( (Vtx ` q ) = V /\ (Edg ` q ) = a ) ) ) )
66	52, 65	mpbird 247	. . . . . . . 8 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> <. V , a >. e. G )
67		fveq2 6191	. . . . . . . . . 10 |- ( b = <. V , a >. -> ( 2nd ` b ) = ( 2nd ` <. V , a >. ) )
68	67	eqeq2d 2632	. . . . . . . . 9 |- ( b = <. V , a >. -> ( a = ( 2nd ` b ) <-> a = ( 2nd ` <. V , a >. ) ) )
69	68	adantl 482	. . . . . . . 8 |- ( ( ( a C_ (Pairs ` V ) /\ V e. W ) /\ b = <. V , a >. ) -> ( a = ( 2nd ` b ) <-> a = ( 2nd ` <. V , a >. ) ) )
70		op2ndg 7181	. . . . . . . . . . 11 |- ( ( V e. W /\ a e. _V ) -> ( 2nd ` <. V , a >. ) = a )
71	10, 70	mpan2 707	. . . . . . . . . 10 |- ( V e. W -> ( 2nd ` <. V , a >. ) = a )
72	71	adantl 482	. . . . . . . . 9 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> ( 2nd ` <. V , a >. ) = a )
73	72	eqcomd 2628	. . . . . . . 8 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> a = ( 2nd ` <. V , a >. ) )
74	66, 69, 73	rspcedvd 3317	. . . . . . 7 |- ( ( a C_ (Pairs ` V ) /\ V e. W ) -> E. b e. G a = ( 2nd ` b ) )
75	74	ex 450	. . . . . 6 |- ( a C_ (Pairs ` V ) -> ( V e. W -> E. b e. G a = ( 2nd ` b ) ) )
76	8, 75	sylbi 207	. . . . 5 |- ( a e. P -> ( V e. W -> E. b e. G a = ( 2nd ` b ) ) )
77	76	impcom 446	. . . 4 |- ( ( V e. W /\ a e. P ) -> E. b e. G a = ( 2nd ` b ) )
78	1, 2, 3	uspgrsprfv 41753	. . . . . . 7 |- ( b e. G -> ( F ` b ) = ( 2nd ` b ) )
79	78	adantl 482	. . . . . 6 |- ( ( ( V e. W /\ a e. P ) /\ b e. G ) -> ( F ` b ) = ( 2nd ` b ) )
80	79	eqeq2d 2632	. . . . 5 |- ( ( ( V e. W /\ a e. P ) /\ b e. G ) -> ( a = ( F ` b ) <-> a = ( 2nd ` b ) ) )
81	80	rexbidva 3049	. . . 4 |- ( ( V e. W /\ a e. P ) -> ( E. b e. G a = ( F ` b ) <-> E. b e. G a = ( 2nd ` b ) ) )
82	77, 81	mpbird 247	. . 3 |- ( ( V e. W /\ a e. P ) -> E. b e. G a = ( F ` b ) )
83	82	ralrimiva 2966	. 2 |- ( V e. W -> A. a e. P E. b e. G a = ( F ` b ) )
84		dffo3 6374	. 2 |- ( F : G -onto-> P <-> ( F : G --> P /\ A. a e. P E. b e. G a = ( F ` b ) ) )
85	5, 83, 84	sylanbrc 698	1 |- ( V e. W -> F : G -onto-> P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   2ndc2nd 7167   <_ cle 10075   2c2 11070   #chash 13117  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-vtx 25876  df-iedg 25877  df-edg 25940  df-upgr 25977  df-uspgr 26045  df-spr 41728

This theorem is referenced by:  uspgrsprf1o  41757