MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem8 Unicode version

Theorem tfrlem8 6416
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 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 6409 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
32abeq2i 2403 . . . . . . 7  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
4 fndm 5359 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  dom  g  =  z )
54adantr 451 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  dom  g  =  z
)
65eleq1d 2362 . . . . . . . . 9  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  -> 
( dom  g  e.  On 
<->  z  e.  On ) )
76biimprcd 216 . . . . . . . 8  |-  ( z  e.  On  ->  (
( g  Fn  z  /\  A. y  e.  z  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) )  ->  dom  g  e.  On ) )
87rexlimiv 2674 . . . . . . 7  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  dom  g  e.  On )
93, 8sylbi 187 . . . . . 6  |-  ( g  e.  A  ->  dom  g  e.  On )
10 eleq1a 2365 . . . . . 6  |-  ( dom  g  e.  On  ->  ( z  =  dom  g  ->  z  e.  On ) )
119, 10syl 15 . . . . 5  |-  ( g  e.  A  ->  (
z  =  dom  g  ->  z  e.  On ) )
1211rexlimiv 2674 . . . 4  |-  ( E. g  e.  A  z  =  dom  g  -> 
z  e.  On )
1312abssi 3261 . . 3  |-  { z  |  E. g  e.  A  z  =  dom  g }  C_  On
14 ssorduni 4593 . . 3  |-  ( { z  |  E. g  e.  A  z  =  dom  g }  C_  On  ->  Ord  U. { z  |  E. g  e.  A  z  =  dom  g } )
1513, 14ax-mp 8 . 2  |-  Ord  U. { z  |  E. g  e.  A  z  =  dom  g }
161recsfval 6413 . . . . 5  |- recs ( F )  =  U. A
1716dmeqi 4896 . . . 4  |-  dom recs ( F )  =  dom  U. A
18 dmuni 4904 . . . 4  |-  dom  U. A  =  U_ g  e.  A  dom  g
19 vex 2804 . . . . . 6  |-  g  e. 
_V
2019dmex 4957 . . . . 5  |-  dom  g  e.  _V
2120dfiun2 3953 . . . 4  |-  U_ g  e.  A  dom  g  = 
U. { z  |  E. g  e.  A  z  =  dom  g }
2217, 18, 213eqtri 2320 . . 3  |-  dom recs ( F )  =  U. { z  |  E. g  e.  A  z  =  dom  g }
23 ordeq 4415 . . 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 8 . 2  |-  ( Ord 
dom recs ( F )  <->  Ord  U. {
z  |  E. g  e.  A  z  =  dom  g } )
2515, 24mpbir 200 1  |-  Ord  dom recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   U_ciun 3921   Ord word 4407   Oncon0 4408   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271  recscrecs 6403
This theorem is referenced by:  tfrlem10  6419  tfrlem12  6421  tfrlem13  6422  tfrlem14  6423  tfrlem15  6424  tfrlem16  6425
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-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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  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-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-recs 6404
  Copyright terms: Public domain W3C validator