Users' Mathboxes Mathbox for Drahflow < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rtrclreclem.subset Unicode version

Theorem rtrclreclem.subset 24917
Description: The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
rtrclreclem.1  |-  ( ph  ->  Rel  R )
rtrclreclem.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
rtrclreclem.subset  |-  ( ph  ->  R  C_  ( t *rec `  R )
)

Proof of Theorem rtrclreclem.subset
Dummy variables  r  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1nn0 10162 . . . . 5  |-  1  e.  NN0
2 ssid 3303 . . . . . . 7  |-  R  C_  R
32a1i 11 . . . . . 6  |-  ( ph  ->  R  C_  R )
4 rtrclreclem.1 . . . . . . 7  |-  ( ph  ->  Rel  R )
5 rtrclreclem.2 . . . . . . 7  |-  ( ph  ->  R  e.  _V )
64, 5relexp1 24903 . . . . . 6  |-  ( ph  ->  ( R ^ r 1 )  =  R )
73, 6sseqtr4d 3321 . . . . 5  |-  ( ph  ->  R  C_  ( R ^ r 1 ) )
8 oveq2 6021 . . . . . . 7  |-  ( n  =  1  ->  ( R ^ r n )  =  ( R ^
r 1 ) )
98sseq2d 3312 . . . . . 6  |-  ( n  =  1  ->  ( R  C_  ( R ^
r n )  <->  R  C_  ( R ^ r 1 ) ) )
109rspcev 2988 . . . . 5  |-  ( ( 1  e.  NN0  /\  R  C_  ( R ^
r 1 ) )  ->  E. n  e.  NN0  R 
C_  ( R ^
r n ) )
111, 7, 10sylancr 645 . . . 4  |-  ( ph  ->  E. n  e.  NN0  R 
C_  ( R ^
r n ) )
12 ssiun 4067 . . . 4  |-  ( E. n  e.  NN0  R  C_  ( R ^ r n )  ->  R  C_ 
U_ n  e.  NN0  ( R ^ r n ) )
1311, 12syl 16 . . 3  |-  ( ph  ->  R  C_  U_ n  e. 
NN0  ( R ^
r n ) )
14 eqidd 2381 . . . 4  |-  ( ph  ->  ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) )  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) )
15 oveq1 6020 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
1615iuneq2d 4053 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
1716adantl 453 . . . 4  |-  ( (
ph  /\  r  =  R )  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
18 nn0ex 10152 . . . . . 6  |-  NN0  e.  _V
19 ovex 6038 . . . . . 6  |-  ( R ^ r n )  e.  _V
2018, 19iunex 5923 . . . . 5  |-  U_ n  e.  NN0  ( R ^
r n )  e. 
_V
2120a1i 11 . . . 4  |-  ( ph  ->  U_ n  e.  NN0  ( R ^ r n )  e.  _V )
2214, 17, 5, 21fvmptd 5742 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
2313, 22sseqtr4d 3321 . 2  |-  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) )
24 df-rtrclrec 24914 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
25 fveq1 5660 . . . . 5  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( t *rec
`  R )  =  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) )
2625sseq2d 3312 . . . 4  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( R  C_  ( t *rec `  R )  <->  R  C_  (
( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) ) `  R
) ) )
2726imbi2d 308 . . 3  |-  ( t *rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )  ->  ( ( ph  ->  R  C_  ( t *rec `  R )
)  <->  ( ph  ->  R 
C_  ( ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) ) `
 R ) ) ) )
2824, 27ax-mp 8 . 2  |-  ( (
ph  ->  R  C_  (
t *rec `  R
) )  <->  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) ) )
2923, 28mpbir 201 1  |-  ( ph  ->  R  C_  ( t *rec `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   E.wrex 2643   _Vcvv 2892    C_ wss 3256   U_ciun 4028    e. cmpt 4200   Rel wrel 4816   ` cfv 5387  (class class class)co 6013   1c1 8917   NN0cn0 10146   ^ rcrelexp 24899   t *reccrtrcl 24913
This theorem is referenced by:  dfrtrcl2  24920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-n0 10147  df-z 10208  df-uz 10414  df-seq 11244  df-relexp 24900  df-rtrclrec 24914
  Copyright terms: Public domain W3C validator