ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem8 Unicode version

Theorem tfrlem8 6183
Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem8  |-  Ord  dom recs ( F )
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem8
Dummy variables  g  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . . . . 9  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem3 6176 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2228 . . . . . . 7  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fndm 5192 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  dom  g  =  z )
54adantr 274 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  dom  g  =  z
)
65eleq1d 2186 . . . . . . . . 9  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  -> 
( dom  g  e.  On 
<->  z  e.  On ) )
76biimprcd 159 . . . . . . . 8  |-  ( z  e.  On  ->  (
( g  Fn  z  /\  A. w  e.  z  ( g `  w
)  =  ( F `
 ( g  |`  w ) ) )  ->  dom  g  e.  On ) )
87rexlimiv 2520 . . . . . . 7  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  dom  g  e.  On )
93, 8sylbi 120 . . . . . 6  |-  ( g  e.  A  ->  dom  g  e.  On )
10 eleq1a 2189 . . . . . 6  |-  ( dom  g  e.  On  ->  ( z  =  dom  g  ->  z  e.  On ) )
119, 10syl 14 . . . . 5  |-  ( g  e.  A  ->  (
z  =  dom  g  ->  z  e.  On ) )
1211rexlimiv 2520 . . . 4  |-  ( E. g  e.  A  z  =  dom  g  -> 
z  e.  On )
1312abssi 3142 . . 3  |-  { z  |  E. g  e.  A  z  =  dom  g }  C_  On
14 ssorduni 4373 . . 3  |-  ( { z  |  E. g  e.  A  z  =  dom  g }  C_  On  ->  Ord  U. { z  |  E. g  e.  A  z  =  dom  g } )
1513, 14ax-mp 5 . 2  |-  Ord  U. { z  |  E. g  e.  A  z  =  dom  g }
161recsfval 6180 . . . . 5  |- recs ( F )  =  U. A
1716dmeqi 4710 . . . 4  |-  dom recs ( F )  =  dom  U. A
18 dmuni 4719 . . . 4  |-  dom  U. A  =  U_ g  e.  A  dom  g
19 vex 2663 . . . . . 6  |-  g  e. 
_V
2019dmex 4775 . . . . 5  |-  dom  g  e.  _V
2120dfiun2 3817 . . . 4  |-  U_ g  e.  A  dom  g  = 
U. { z  |  E. g  e.  A  z  =  dom  g }
2217, 18, 213eqtri 2142 . . 3  |-  dom recs ( F )  =  U. { z  |  E. g  e.  A  z  =  dom  g }
23 ordeq 4264 . . 3  |-  ( dom recs
( F )  = 
U. { z  |  E. g  e.  A  z  =  dom  g }  ->  ( Ord  dom recs ( F )  <->  Ord  U. {
z  |  E. g  e.  A  z  =  dom  g } ) )
2422, 23ax-mp 5 . 2  |-  ( Ord 
dom recs ( F )  <->  Ord  U. {
z  |  E. g  e.  A  z  =  dom  g } )
2515, 24mpbir 145 1  |-  Ord  dom recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394    C_ wss 3041   U.cuni 3706   U_ciun 3783   Ord word 4254   Oncon0 4255   dom cdm 4509    |` cres 4511    Fn wfn 5088   ` cfv 5093  recscrecs 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-tr 3997  df-iord 4258  df-on 4260  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-recs 6170
This theorem is referenced by:  tfrlemi14d  6198  tfri1dALT  6216
  Copyright terms: Public domain W3C validator