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Theorem usgr2pth0 26661
Description: In a simply graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
Hypotheses
Ref Expression
usgr2pthlem.v |- V = (Vtx ` G )
usgr2pthlem.i |- I = (iEdg ` G )
Assertion
Ref Expression
usgr2pth0 |- ( G e. USGraph -> ( ( F (Paths ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
Distinct variable groups:   x, F, y, z   x, G, y, z   x, I, y, z   x, P, y, z   x, V, y, z
Proof of Theorem usgr2pth0
StepHypRef Expression
1 usgr2pthlem.v . . 3 |- V = (Vtx ` G )
2 usgr2pthlem.i . . 3 |- I = (iEdg ` G )
31, 2usgr2pth 26660 . 2 |- ( G e. USGraph -> ( ( F (Paths ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
4 r19.42v 3092 . . . . . . . . 9 |- ( E. y e. ( V \ { x , z } ) ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( z =/= x /\ E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
5 rexdifpr 4205 . . . . . . . . 9 |- ( E. y e. ( V \ { x , z } ) ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> E. y e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
64, 5bitr3i 266 . . . . . . . 8 |- ( ( z =/= x /\ E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> E. y e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
76rexbii 3041 . . . . . . 7 |- ( E. z e. V ( z =/= x /\ E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> E. z e. V E. y e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
8 rexcom 3099 . . . . . . 7 |- ( E. z e. V E. y e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> E. y e. V E. z e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
9 df-3an 1039 . . . . . . . . . . 11 |- ( ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> ( ( y =/= x /\ y =/= z ) /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
10 anass 681 . . . . . . . . . . 11 |- ( ( ( ( y =/= x /\ y =/= z ) /\ z =/= x ) /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( ( y =/= x /\ y =/= z ) /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
11 anass 681 . . . . . . . . . . . 12 |- ( ( ( ( z =/= x /\ z =/= y ) /\ y =/= x ) /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( ( z =/= x /\ z =/= y ) /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
12 anass 681 . . . . . . . . . . . . . 14 |- ( ( ( y =/= x /\ y =/= z ) /\ z =/= x ) <-> ( y =/= x /\ ( y =/= z /\ z =/= x ) ) )
13 ancom 466 . . . . . . . . . . . . . 14 |- ( ( y =/= x /\ ( y =/= z /\ z =/= x ) ) <-> ( ( y =/= z /\ z =/= x ) /\ y =/= x ) )
14 necom 2847 . . . . . . . . . . . . . . . 16 |- ( y =/= z <-> z =/= y )
1514anbi2ci 732 . . . . . . . . . . . . . . 15 |- ( ( y =/= z /\ z =/= x ) <-> ( z =/= x /\ z =/= y ) )
1615anbi1i 731 . . . . . . . . . . . . . 14 |- ( ( ( y =/= z /\ z =/= x ) /\ y =/= x ) <-> ( ( z =/= x /\ z =/= y ) /\ y =/= x ) )
1712, 13, 163bitri 286 . . . . . . . . . . . . 13 |- ( ( ( y =/= x /\ y =/= z ) /\ z =/= x ) <-> ( ( z =/= x /\ z =/= y ) /\ y =/= x ) )
1817anbi1i 731 . . . . . . . . . . . 12 |- ( ( ( ( y =/= x /\ y =/= z ) /\ z =/= x ) /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( ( ( z =/= x /\ z =/= y ) /\ y =/= x ) /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
19 df-3an 1039 . . . . . . . . . . . 12 |- ( ( z =/= x /\ z =/= y /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> ( ( z =/= x /\ z =/= y ) /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
2011, 18, 193bitr4i 292 . . . . . . . . . . 11 |- ( ( ( ( y =/= x /\ y =/= z ) /\ z =/= x ) /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( z =/= x /\ z =/= y /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
219, 10, 203bitr2i 288 . . . . . . . . . 10 |- ( ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> ( z =/= x /\ z =/= y /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
2221rexbii 3041 . . . . . . . . 9 |- ( E. z e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> E. z e. V ( z =/= x /\ z =/= y /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
23 rexdifpr 4205 . . . . . . . . 9 |- ( E. z e. ( V \ { x , y } ) ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> E. z e. V ( z =/= x /\ z =/= y /\ ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
24 r19.42v 3092 . . . . . . . . 9 |- ( E. z e. ( V \ { x , y } ) ( y =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( y =/= x /\ E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
2522, 23, 243bitr2i 288 . . . . . . . 8 |- ( E. z e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> ( y =/= x /\ E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
2625rexbii 3041 . . . . . . 7 |- ( E. y e. V E. z e. V ( y =/= x /\ y =/= z /\ ( z =/= x /\ ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) <-> E. y e. V ( y =/= x /\ E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
277, 8, 263bitri 286 . . . . . 6 |- ( E. z e. V ( z =/= x /\ E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> E. y e. V ( y =/= x /\ E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
28 rexdifsn 4323 . . . . . 6 |- ( E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) <-> E. z e. V ( z =/= x /\ E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
29 rexdifsn 4323 . . . . . 6 |- ( E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) <-> E. y e. V ( y =/= x /\ E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
3027, 28, 293bitr4i 292 . . . . 5 |- ( E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) <-> E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) )
3130a1i 11 . . . 4 |- ( ( G e. USGraph /\ x e. V ) -> ( E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) <-> E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
3231rexbidva 3049 . . 3 |- ( G e. USGraph -> ( E. x e. V E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) <-> E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) )
33323anbi3d 1405 . 2 |- ( G e. USGraph -> ( ( F : ( 0..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. z e. ( V \ { x } ) E. y e. ( V \ { x , z } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) <-> ( F : ( 0..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
343, 33bitrd 268 1 |- ( G e. USGraph -> ( ( F (Paths ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = z /\ ( P ` 2 ) = y ) /\ ( ( I ` ( F ` 0 ) ) = { x , z } /\ ( I ` ( F ` 1 ) ) = { z , y } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   USGraph cusgr 26044  Pathscpths 26608
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-ifp 1013  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-map 7859  df-pm 7860  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-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615
This theorem is referenced by: (None)
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