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Theorem rtrclreclem.subset 28819
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 10812 . . . . 5  |-  1  e.  NN0
2 ssid 3523 . . . . . . 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 28805 . . . . . 6  |-  ( ph  ->  ( R ^r 1 )  =  R )
73, 6sseqtr4d 3541 . . . . 5  |-  ( ph  ->  R  C_  ( R ^r 1 ) )
8 oveq2 6293 . . . . . . 7  |-  ( n  =  1  ->  ( R ^r n )  =  ( R ^r 1 ) )
98sseq2d 3532 . . . . . 6  |-  ( n  =  1  ->  ( R  C_  ( R ^r n )  <->  R  C_  ( R ^r 1
) ) )
109rspcev 3214 . . . . 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 4367 . . . 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 2468 . . . 4  |-  ( ph  ->  ( r  e.  _V  |->  U_ n  e.  NN0  (
r ^r n ) )  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) ) )
15 oveq1 6292 . . . . . 6  |-  ( r  =  R  ->  (
r ^r n )  =  ( R ^r n ) )
1615iuneq2d 4352 . . . . 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 10802 . . . . . 6  |-  NN0  e.  _V
19 ovex 6310 . . . . . 6  |-  ( R ^r n )  e.  _V
2018, 19iunex 6765 . . . . 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 5956 . . 3  |-  ( ph  ->  ( ( r  e. 
_V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R )  =  U_ n  e.  NN0  ( R ^r n ) )
2313, 22sseqtr4d 3541 . 2  |-  ( ph  ->  R  C_  ( (
r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) ) `  R ) )
24 df-rtrclrec 28816 . . 3  |-  t*rec  =  ( r  e.  _V  |->  U_ n  e.  NN0  ( r ^r n ) )
25 fveq1 5865 . . . . 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 3532 . . . 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 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476   U_ciun 4325    |-> cmpt 4505   Rel wrel 5004   ` cfv 5588  (class class class)co 6285   1c1 9494   NN0cn0 10796   ^rcrelexp 28801   t*reccrtrcl 28815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-seq 12077  df-relexp 28802  df-rtrclrec 28816
This theorem is referenced by:  dfrtrcl2  28822
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