MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1lim Unicode version

Theorem r1lim 7534
Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1lim  |-  ( ( A  e.  B  /\  Lim  A )  ->  ( R1 `  A )  = 
U_ x  e.  A  ( R1 `  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem r1lim
StepHypRef Expression
1 limelon 4537 . . 3  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
2 r1fnon 7529 . . . 4  |-  R1  Fn  On
3 fndm 5425 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
42, 3ax-mp 8 . . 3  |-  dom  R1  =  On
51, 4syl6eleqr 2449 . 2  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  dom  R1 )
6 r1limg 7533 . 2  |-  ( ( A  e.  dom  R1  /\ 
Lim  A )  -> 
( R1 `  A
)  =  U_ x  e.  A  ( R1 `  x ) )
75, 6sylancom 648 1  |-  ( ( A  e.  B  /\  Lim  A )  ->  ( R1 `  A )  = 
U_ x  e.  A  ( R1 `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   U_ciun 3986   Oncon0 4474   Lim wlim 4475   dom cdm 4771    Fn wfn 5332   ` cfv 5337   R1cr1 7524
This theorem is referenced by:  r1sdom  7536  r1om  7960  inar1  8487  inatsk  8490  grur1a  8531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-r1 7526
  Copyright terms: Public domain W3C validator