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Theorem rtrclreclem.subset 27294
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 10587 . . . . 5  |-  1  e.  NN0
2 ssid 3368 . . . . . . 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 27280 . . . . . 6  |-  ( ph  ->  ( R ^r 1 )  =  R )
73, 6sseqtr4d 3386 . . . . 5  |-  ( ph  ->  R  C_  ( R ^r 1 ) )
8 oveq2 6094 . . . . . . 7  |-  ( n  =  1  ->  ( R ^r n )  =  ( R ^r 1 ) )
98sseq2d 3377 . . . . . 6  |-  ( n  =  1  ->  ( R  C_  ( R ^r n )  <->  R  C_  ( R ^r 1
) ) )
109rspcev 3066 . . . . 5  |-  ( ( 1  e.  NN0  /\  R  C_  ( R ^r 1 ) )  ->  E. n  e.  NN0  R 
C_  ( R ^r n ) )
111, 7, 10sylancr 663 . . . 4  |-  ( ph  ->  E. n  e.  NN0  R 
C_  ( R ^r n ) )
12 ssiun 4205 . . . 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 2438 . . . 4  |-  ( ph  ->  ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^r n ) )  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) ) )
15 oveq1 6093 . . . . . 6  |-  ( r  =  R  ->  (
r ^r n )  =  ( R ^r n ) )
1615iuneq2d 4190 . . . . 5  |-  ( r  =  R  ->  U_ n  e.  NN0  ( r ^r n )  = 
U_ n  e.  NN0  ( R ^r n ) )
1716adantl 466 . . . 4  |-  ( (
ph  /\  r  =  R )  ->  U_ n  e.  NN0  ( r ^r n )  = 
U_ n  e.  NN0  ( R ^r n ) )
18 nn0ex 10577 . . . . . 6  |-  NN0  e.  _V
19 ovex 6111 . . . . . 6  |-  ( R ^r n )  e.  _V
2018, 19iunex 6552 . . . . 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 5772 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^r n ) )
2313, 22sseqtr4d 3386 . 2  |-  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R ) )
24 df-rtrclrec 27291 . . 3  |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )
25 fveq1 5683 . . . . 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 3377 . . . 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 316 . . 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 5 . 2  |-  ( (
ph  ->  R  C_  (
t*rec `  R
) )  <->  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R ) ) )
2923, 28mpbir 209 1  |-  ( ph  ->  R  C_  ( t*rec `  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   E.wrex 2710   _Vcvv 2966    C_ wss 3321   U_ciun 4164    e. cmpt 4343   Rel wrel 4837   ` cfv 5411  (class class class)co 6086   1c1 9275   NN0cn0 10571   ^rcrelexp 27276   t*reccrtrcl 27290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-seq 11799  df-relexp 27277  df-rtrclrec 27291
This theorem is referenced by:  dfrtrcl2  27297
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