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Theorem rtrclreclem.subset 24042
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 9981 . . . . 5  |-  1  e.  NN0
2 ssid 3197 . . . . . . 7  |-  R  C_  R
32a1i 10 . . . . . 6  |-  ( ph  ->  R  C_  R )
4 rtrclreclem.1 . . . . . . 7  |-  ( ph  ->  Rel  R )
5 rtrclreclem.2 . . . . . . 7  |-  ( ph  ->  R  e.  _V )
64, 5relexp1 24027 . . . . . 6  |-  ( ph  ->  ( R ^ r 1 )  =  R )
73, 6sseqtr4d 3215 . . . . 5  |-  ( ph  ->  R  C_  ( R ^ r 1 ) )
8 oveq2 5866 . . . . . . 7  |-  ( n  =  1  ->  ( R ^ r n )  =  ( R ^
r 1 ) )
98sseq2d 3206 . . . . . 6  |-  ( n  =  1  ->  ( R  C_  ( R ^
r n )  <->  R  C_  ( R ^ r 1 ) ) )
109rspcev 2884 . . . . 5  |-  ( ( 1  e.  NN0  /\  R  C_  ( R ^
r 1 ) )  ->  E. n  e.  NN0  R 
C_  ( R ^
r n ) )
111, 7, 10sylancr 644 . . . 4  |-  ( ph  ->  E. n  e.  NN0  R 
C_  ( R ^
r n ) )
12 ssiun 3944 . . . 4  |-  ( E. n  e.  NN0  R  C_  ( R ^ r n )  ->  R  C_ 
U_ n  e.  NN0  ( R ^ r n ) )
1311, 12syl 15 . . 3  |-  ( ph  ->  R  C_  U_ n  e. 
NN0  ( R ^
r n ) )
14 eqidd 2284 . . . 4  |-  ( ph  ->  ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) )  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) )
15 oveq1 5865 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
1615iuneq2d 3930 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
1716adantl 452 . . . 4  |-  ( (
ph  /\  r  =  R )  ->  U_ n  e.  NN0  ( r ^
r n )  = 
U_ n  e.  NN0  ( R ^ r n ) )
18 nn0ex 9971 . . . . . 6  |-  NN0  e.  _V
19 ovex 5883 . . . . . 6  |-  ( R ^ r n )  e.  _V
2018, 19iunex 5770 . . . . 5  |-  U_ n  e.  NN0  ( R ^
r n )  e. 
_V
2120a1i 10 . . . 4  |-  ( ph  ->  U_ n  e.  NN0  ( R ^ r n )  e.  _V )
2214, 17, 5, 21fvmptd 5606 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
2313, 22sseqtr4d 3215 . 2  |-  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) )
24 df-rtrclrec 24039 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
25 fveq1 5524 . . . . 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 3206 . . . 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 307 . . 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 200 1  |-  ( ph  ->  R  C_  ( t *rec `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U_ciun 3905    e. cmpt 4077   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1c1 8738   NN0cn0 9965   ^ rcrelexp 24023   t *reccrtrcl 24038
This theorem is referenced by:  dfrtrcl2  24045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-relexp 24024  df-rtrclrec 24039
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