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Theorem r1limg 7459
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 7452 . . . . 5  |-  R1  =  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )
21dmeqi 4896 . . . 4  |-  dom  R1  =  dom  rec ( ( y  e.  _V  |->  ~P y ) ,  (/) )
32eleq2i 2360 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  dom  rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) )
4 rdglimg 6454 . . 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 5542 . 2  |-  ( R1
`  A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )
7 r1funlim 7454 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
87simpli 444 . . . 4  |-  Fun  R1
9 funiunfv 5790 . . . 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 5025 . . . 4  |-  ( R1
" A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1211unieqi 3853 . . 3  |-  U. ( R1 " A )  = 
U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1310, 12eqtri 2316 . 2  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) " A )
145, 6, 133eqtr4g 2353 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   U_ciun 3921    e. cmpt 4093   Lim wlim 4409   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271   reccrdg 6438   R1cr1 7450
This theorem is referenced by:  r1lim  7460  r1tr  7464  r1ordg  7466  r1pwss  7472  r1val1  7474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452
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