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Theorem tfrlem8 6083
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 6076 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2198 . . . . . . 7  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fndm 5113 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  dom  g  =  z )
54adantr 270 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  dom  g  =  z
)
65eleq1d 2156 . . . . . . . . 9  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  -> 
( dom  g  e.  On 
<->  z  e.  On ) )
76biimprcd 158 . . . . . . . 8  |-  ( z  e.  On  ->  (
( g  Fn  z  /\  A. w  e.  z  ( g `  w
)  =  ( F `
 ( g  |`  w ) ) )  ->  dom  g  e.  On ) )
87rexlimiv 2483 . . . . . . 7  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  dom  g  e.  On )
93, 8sylbi 119 . . . . . 6  |-  ( g  e.  A  ->  dom  g  e.  On )
10 eleq1a 2159 . . . . . 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 2483 . . . 4  |-  ( E. g  e.  A  z  =  dom  g  -> 
z  e.  On )
1312abssi 3096 . . 3  |-  { z  |  E. g  e.  A  z  =  dom  g }  C_  On
14 ssorduni 4304 . . 3  |-  ( { z  |  E. g  e.  A  z  =  dom  g }  C_  On  ->  Ord  U. { z  |  E. g  e.  A  z  =  dom  g } )
1513, 14ax-mp 7 . 2  |-  Ord  U. { z  |  E. g  e.  A  z  =  dom  g }
161recsfval 6080 . . . . 5  |- recs ( F )  =  U. A
1716dmeqi 4637 . . . 4  |-  dom recs ( F )  =  dom  U. A
18 dmuni 4646 . . . 4  |-  dom  U. A  =  U_ g  e.  A  dom  g
19 vex 2622 . . . . . 6  |-  g  e. 
_V
2019dmex 4699 . . . . 5  |-  dom  g  e.  _V
2120dfiun2 3764 . . . 4  |-  U_ g  e.  A  dom  g  = 
U. { z  |  E. g  e.  A  z  =  dom  g }
2217, 18, 213eqtri 2112 . . 3  |-  dom recs ( F )  =  U. { z  |  E. g  e.  A  z  =  dom  g }
23 ordeq 4199 . . 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 7 . 2  |-  ( Ord 
dom recs ( F )  <->  Ord  U. {
z  |  E. g  e.  A  z  =  dom  g } )
2515, 24mpbir 144 1  |-  Ord  dom recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360    C_ wss 2999   U.cuni 3653   U_ciun 3730   Ord word 4189   Oncon0 4190   dom cdm 4438    |` cres 4440    Fn wfn 5010   ` cfv 5015  recscrecs 6069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-tr 3937  df-iord 4193  df-on 4195  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-recs 6070
This theorem is referenced by:  tfrlemi14d  6098  tfri1dALT  6116
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