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Theorem r1limg 7445
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 7438 . . . . 5  |-  R1  =  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )
21dmeqi 4882 . . . 4  |-  dom  R1  =  dom  rec ( ( y  e.  _V  |->  ~P y ) ,  (/) )
32eleq2i 2349 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  dom  rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) )
4 rdglimg 6440 . . 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 5528 . 2  |-  ( R1
`  A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )
7 r1funlim 7440 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
87simpli 444 . . . 4  |-  Fun  R1
9 funiunfv 5776 . . . 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 5011 . . . 4  |-  ( R1
" A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1211unieqi 3839 . . 3  |-  U. ( R1 " A )  = 
U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1310, 12eqtri 2305 . 2  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) " A )
145, 6, 133eqtr4g 2342 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 1625    e. wcel 1686   _Vcvv 2790   (/)c0 3457   ~Pcpw 3627   U.cuni 3829   U_ciun 3907    e. cmpt 4079   Lim wlim 4395   dom cdm 4691   "cima 4694   Fun wfun 5251   ` cfv 5257   reccrdg 6424   R1cr1 7436
This theorem is referenced by:  r1lim  7446  r1tr  7450  r1ordg  7452  r1pwss  7458  r1val1  7460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438
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