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Theorem rtrclreclem.subset 24057
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 9997 . . . . 5  |-  1  e.  NN0
2 ssid 3210 . . . . . . 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 24042 . . . . . 6  |-  ( ph  ->  ( R ^ r 1 )  =  R )
73, 6sseqtr4d 3228 . . . . 5  |-  ( ph  ->  R  C_  ( R ^ r 1 ) )
8 oveq2 5882 . . . . . . 7  |-  ( n  =  1  ->  ( R ^ r n )  =  ( R ^
r 1 ) )
98sseq2d 3219 . . . . . 6  |-  ( n  =  1  ->  ( R  C_  ( R ^
r n )  <->  R  C_  ( R ^ r 1 ) ) )
109rspcev 2897 . . . . 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 3960 . . . 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 2297 . . . 4  |-  ( ph  ->  ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^ r n ) )  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) )
15 oveq1 5881 . . . . . 6  |-  ( r  =  R  ->  (
r ^ r n )  =  ( R ^ r n ) )
1615iuneq2d 3946 . . . . 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 9987 . . . . . 6  |-  NN0  e.  _V
19 ovex 5899 . . . . . 6  |-  ( R ^ r n )  e.  _V
2018, 19iunex 5786 . . . . 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 5622 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^ r n ) )
2313, 22sseqtr4d 3228 . 2  |-  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^ r n ) ) `  R ) )
24 df-rtrclrec 24054 . . 3  |-  t
*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^
r n ) )
25 fveq1 5540 . . . . 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 3219 . . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   U_ciun 3921    e. cmpt 4093   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   1c1 8754   NN0cn0 9981   ^ rcrelexp 24038   t *reccrtrcl 24053
This theorem is referenced by:  dfrtrcl2  24060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-relexp 24039  df-rtrclrec 24054
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