MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1limg Structured version   Unicode version

Theorem r1limg 8178
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 8171 . . . . 5  |-  R1  =  rec ( ( y  e. 
_V  |->  ~P y ) ,  (/) )
21dmeqi 5195 . . . 4  |-  dom  R1  =  dom  rec ( ( y  e.  _V  |->  ~P y ) ,  (/) )
32eleq2i 2538 . . 3  |-  ( A  e.  dom  R1  <->  A  e.  dom  rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) )
4 rdglimg 7081 . . 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 472 . 2  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )  =  U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A ) )
61fveq1i 5858 . 2  |-  ( R1
`  A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) `  A )
7 r1funlim 8173 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
87simpli 458 . . . 4  |-  Fun  R1
9 funiunfv 6139 . . . 4  |-  ( Fun 
R1  ->  U_ x  e.  A  ( R1 `  x )  =  U. ( R1
" A ) )
108, 9ax-mp 5 . . 3  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( R1 " A )
111imaeq1i 5325 . . . 4  |-  ( R1
" A )  =  ( rec ( ( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1211unieqi 4247 . . 3  |-  U. ( R1 " A )  = 
U. ( rec (
( y  e.  _V  |->  ~P y ) ,  (/) ) " A )
1310, 12eqtri 2489 . 2  |-  U_ x  e.  A  ( R1 `  x )  =  U. ( rec ( ( y  e.  _V  |->  ~P y
) ,  (/) ) " A )
145, 6, 133eqtr4g 2526 1  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   U_ciun 4318    |-> cmpt 4498   Lim wlim 4872   dom cdm 4992   "cima 4995   Fun wfun 5573   ` cfv 5579   reccrdg 7065   R1cr1 8169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-recs 7032  df-rdg 7066  df-r1 8171
This theorem is referenced by:  r1lim  8179  r1tr  8183  r1ordg  8185  r1pwss  8191  r1val1  8193
  Copyright terms: Public domain W3C validator