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Theorem tfrlem8 5967
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 5960 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2190 . . . . . . 7  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fndm 5029 . . . . . . . . . . 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 2148 . . . . . . . . 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 2472 . . . . . . 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 2151 . . . . . 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 2472 . . . 4  |-  ( E. g  e.  A  z  =  dom  g  -> 
z  e.  On )
1312abssi 3070 . . 3  |-  { z  |  E. g  e.  A  z  =  dom  g }  C_  On
14 ssorduni 4239 . . 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 5964 . . . . 5  |- recs ( F )  =  U. A
1716dmeqi 4564 . . . 4  |-  dom recs ( F )  =  dom  U. A
18 dmuni 4573 . . . 4  |-  dom  U. A  =  U_ g  e.  A  dom  g
19 vex 2605 . . . . . 6  |-  g  e. 
_V
2019dmex 4626 . . . . 5  |-  dom  g  e.  _V
2120dfiun2 3720 . . . 4  |-  U_ g  e.  A  dom  g  = 
U. { z  |  E. g  e.  A  z  =  dom  g }
2217, 18, 213eqtri 2106 . . 3  |-  dom recs ( F )  =  U. { z  |  E. g  e.  A  z  =  dom  g }
23 ordeq 4135 . . 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 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350    C_ wss 2974   U.cuni 3609   U_ciun 3686   Ord word 4125   Oncon0 4126   dom cdm 4371    |` cres 4373    Fn wfn 4927   ` cfv 4932  recscrecs 5953
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-tr 3884  df-iord 4129  df-on 4131  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954
This theorem is referenced by:  tfrlemi14d  5982  tfri1dALT  6000
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