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Theorem r1limg 7631
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 7624 . . . . 5  |-  R1  =  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )
21dmeqi 5012 . . . 4  |-  dom  R1  =  dom  rec ( ( y  e.  _V  |->  ~P y ) ,  (/) )
32eleq2i 2452 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  dom  rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) )
4 rdglimg 6620 . . 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 459 . 2  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )  =  U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A ) )
61fveq1i 5670 . 2  |-  ( R1
`  A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )
7 r1funlim 7626 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
87simpli 445 . . . 4  |-  Fun  R1
9 funiunfv 5935 . . . 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 5141 . . . 4  |-  ( R1
" A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1211unieqi 3968 . . 3  |-  U. ( R1 " A )  = 
U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1310, 12eqtri 2408 . 2  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) " A )
145, 6, 133eqtr4g 2445 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 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   (/)c0 3572   ~Pcpw 3743   U.cuni 3958   U_ciun 4036    e. cmpt 4208   Lim wlim 4524   dom cdm 4819   "cima 4822   Fun wfun 5389   ` cfv 5395   reccrdg 6604   R1cr1 7622
This theorem is referenced by:  r1lim  7632  r1tr  7636  r1ordg  7638  r1pwss  7644  r1val1  7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-r1 7624
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