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Theorem tfrlem8 6647
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 6640 . . . . . . . 8  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
32abeq2i 2545 . . . . . . 7  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
4 fndm 5546 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  dom  g  =  z )
54adantr 453 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  dom  g  =  z
)
65eleq1d 2504 . . . . . . . . 9  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  -> 
( dom  g  e.  On 
<->  z  e.  On ) )
76biimprcd 218 . . . . . . . 8  |-  ( z  e.  On  ->  (
( g  Fn  z  /\  A. y  e.  z  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) )  ->  dom  g  e.  On ) )
87rexlimiv 2826 . . . . . . 7  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  dom  g  e.  On )
93, 8sylbi 189 . . . . . 6  |-  ( g  e.  A  ->  dom  g  e.  On )
10 eleq1a 2507 . . . . . 6  |-  ( dom  g  e.  On  ->  ( z  =  dom  g  ->  z  e.  On ) )
119, 10syl 16 . . . . 5  |-  ( g  e.  A  ->  (
z  =  dom  g  ->  z  e.  On ) )
1211rexlimiv 2826 . . . 4  |-  ( E. g  e.  A  z  =  dom  g  -> 
z  e.  On )
1312abssi 3420 . . 3  |-  { z  |  E. g  e.  A  z  =  dom  g }  C_  On
14 ssorduni 4768 . . 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 6644 . . . . 5  |- recs ( F )  =  U. A
1716dmeqi 5073 . . . 4  |-  dom recs ( F )  =  dom  U. A
18 dmuni 5081 . . . 4  |-  dom  U. A  =  U_ g  e.  A  dom  g
19 vex 2961 . . . . . 6  |-  g  e. 
_V
2019dmex 5134 . . . . 5  |-  dom  g  e.  _V
2120dfiun2 4127 . . . 4  |-  U_ g  e.  A  dom  g  = 
U. { z  |  E. g  e.  A  z  =  dom  g }
2217, 18, 213eqtri 2462 . . 3  |-  dom recs ( F )  =  U. { z  |  E. g  e.  A  z  =  dom  g }
23 ordeq 4590 . . 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 202 1  |-  Ord  dom recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    C_ wss 3322   U.cuni 4017   U_ciun 4095   Ord word 4582   Oncon0 4583   dom cdm 4880    |` cres 4882    Fn wfn 5451   ` cfv 5456  recscrecs 6634
This theorem is referenced by:  tfrlem10  6650  tfrlem12  6652  tfrlem13  6653  tfrlem14  6654  tfrlem15  6655  tfrlem16  6656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-recs 6635
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