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Theorem r1limg 7439
Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1limg  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
Distinct variable group:    x, A

Proof of Theorem r1limg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-r1 7432 . . . . 5  |-  R1  =  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )
21dmeqi 4879 . . . 4  |-  dom  R1  =  dom  rec ( ( y  e.  _V  |->  ~P y ) ,  (/) )
32eleq2i 2348 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  dom  rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) )
4 rdglimg 6434 . . 3  |-  ( ( A  e.  dom  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )  /\  Lim  A
)  ->  ( rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) ) `  A )  =  U. ( rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) ) " A ) )
53, 4sylanb 458 . 2  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )  =  U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A ) )
61fveq1i 5487 . 2  |-  ( R1
`  A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )
7 r1funlim 7434 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
87simpli 444 . . . 4  |-  Fun  R1
9 funiunfv 5736 . . . 4  |-  ( Fun 
R1  ->  U_ x  e.  A  ( R1 `  x )  =  U. ( R1
" A ) )
108, 9ax-mp 8 . . 3  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( R1 " A )
111imaeq1i 5008 . . . 4  |-  ( R1
" A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1211unieqi 3838 . . 3  |-  U. ( R1 " A )  = 
U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1310, 12eqtri 2304 . 2  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) " A )
145, 6, 133eqtr4g 2341 1  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   _Vcvv 2789   (/)c0 3456   ~Pcpw 3626   U.cuni 3828   U_ciun 3906    e. cmpt 4078   Lim wlim 4392    dom cdm 4688   "cima 4691   Fun wfun 5215   ` cfv 5221   reccrdg 6418   R1cr1 7430
This theorem is referenced by:  r1lim  7440  r1tr  7444  r1ordg  7446  r1pwss  7452  r1val1  7454
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432
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