Theorem uspgrsprf 41754

Description: The mapping F is a 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, 24-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
uspgrsprf	|- F : G --> 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 uspgrsprf

Step	Hyp	Ref	Expression
1		uspgrsprf.f	. 2 |- F = ( g e. G |-> ( 2nd ` g ) )
2		uspgrsprf.g	. . . . 5 |- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) }
3	2	eleq2i 2693	. . . 4 |- ( g e. G <-> g e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } )
4		elopab 4983	. . . 4 |- ( g e. { <. v , e >. | ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) } <-> E. v E. e ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) )
5	3, 4	bitri 264	. . 3 |- ( g e. G <-> E. v E. e ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) )
6		uspgrupgr 26071	. . . . . . . . . . . . 13 |- ( q e. USPGraph -> q e. UPGraph )
7		upgredgssspr 41751	. . . . . . . . . . . . 13 |- ( q e. UPGraph -> (Edg ` q ) C_ (Pairs ` (Vtx ` q ) ) )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( q e. USPGraph -> (Edg ` q ) C_ (Pairs ` (Vtx ` q ) ) )
9	8	adantr 481	. . . . . . . . . . 11 |- ( ( q e. USPGraph /\ ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> (Edg ` q ) C_ (Pairs ` (Vtx ` q ) ) )
10		simpr 477	. . . . . . . . . . . . 13 |- ( ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) -> (Edg ` q ) = e )
11		fveq2 6191	. . . . . . . . . . . . . 14 |- ( (Vtx ` q ) = v -> (Pairs ` (Vtx ` q ) ) = (Pairs ` v ) )
12	11	adantr 481	. . . . . . . . . . . . 13 |- ( ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) -> (Pairs ` (Vtx ` q ) ) = (Pairs ` v ) )
13	10, 12	sseq12d 3634	. . . . . . . . . . . 12 |- ( ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) -> ( (Edg ` q ) C_ (Pairs ` (Vtx ` q ) ) <-> e C_ (Pairs ` v ) ) )
14	13	adantl 482	. . . . . . . . . . 11 |- ( ( q e. USPGraph /\ ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> ( (Edg ` q ) C_ (Pairs ` (Vtx ` q ) ) <-> e C_ (Pairs ` v ) ) )
15	9, 14	mpbid 222	. . . . . . . . . 10 |- ( ( q e. USPGraph /\ ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> e C_ (Pairs ` v ) )
16	15	rexlimiva 3028	. . . . . . . . 9 |- ( E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) -> e C_ (Pairs ` v ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> e C_ (Pairs ` v ) )
18		fveq2 6191	. . . . . . . . . 10 |- ( v = V -> (Pairs ` v ) = (Pairs ` V ) )
19	18	sseq2d 3633	. . . . . . . . 9 |- ( v = V -> ( e C_ (Pairs ` v ) <-> e C_ (Pairs ` V ) ) )
20	19	adantr 481	. . . . . . . 8 |- ( ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> ( e C_ (Pairs ` v ) <-> e C_ (Pairs ` V ) ) )
21	17, 20	mpbid 222	. . . . . . 7 |- ( ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) -> e C_ (Pairs ` V ) )
22	21	adantl 482	. . . . . 6 |- ( ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> e C_ (Pairs ` V ) )
23		vex 3203	. . . . . . . . 9 |- v e. _V
24		vex 3203	. . . . . . . . 9 |- e e. _V
25	23, 24	op2ndd 7179	. . . . . . . 8 |- ( g = <. v , e >. -> ( 2nd ` g ) = e )
26	25	sseq1d 3632	. . . . . . 7 |- ( g = <. v , e >. -> ( ( 2nd ` g ) C_ (Pairs ` V ) <-> e C_ (Pairs ` V ) ) )
27	26	adantr 481	. . . . . 6 |- ( ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( ( 2nd ` g ) C_ (Pairs ` V ) <-> e C_ (Pairs ` V ) ) )
28	22, 27	mpbird 247	. . . . 5 |- ( ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( 2nd ` g ) C_ (Pairs ` V ) )
29		uspgrsprf.p	. . . . . . 7 |- P = ~P (Pairs ` V )
30	29	eleq2i 2693	. . . . . 6 |- ( ( 2nd ` g ) e. P <-> ( 2nd ` g ) e. ~P (Pairs ` V ) )
31		fvex 6201	. . . . . . 7 |- ( 2nd ` g ) e. _V
32	31	elpw 4164	. . . . . 6 |- ( ( 2nd ` g ) e. ~P (Pairs ` V ) <-> ( 2nd ` g ) C_ (Pairs ` V ) )
33	30, 32	bitri 264	. . . . 5 |- ( ( 2nd ` g ) e. P <-> ( 2nd ` g ) C_ (Pairs ` V ) )
34	28, 33	sylibr 224	. . . 4 |- ( ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( 2nd ` g ) e. P )
35	34	exlimivv 1860	. . 3 |- ( E. v E. e ( g = <. v , e >. /\ ( v = V /\ E. q e. USPGraph ( (Vtx ` q ) = v /\ (Edg ` q ) = e ) ) ) -> ( 2nd ` g ) e. P )
36	5, 35	sylbi 207	. 2 |- ( g e. G -> ( 2nd ` g ) e. P )
37	1, 36	fmpti 6383	1 |- F : G --> P

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   <.cop 4183   {copab 4712   |-> cmpt 4729   -->wf 5884   ` cfv 5888   2ndc2nd 7167  Vtxcvtx 25874  Edgcedg 25939   UPGraph cupgr 25975   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:  uspgrsprf1  41755  uspgrsprfo  41756