MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem7 Unicode version

Theorem tfrlem7 6394
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
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
tfrlem7  |-  Fun recs ( F )
Distinct variable group:    x, f, y, F
Dummy variables  g  h  u  v are mutually distinct and distinct from all other variables.
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem7
StepHypRef Expression
1 tfrlem.1 . . 3  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem6 6393 . 2  |-  Rel recs ( F )
31recsfval 6392 . . . . . . . . 9  |- recs ( F )  =  U. A
43eleq2i 2348 . . . . . . . 8  |-  ( <.
x ,  u >.  e. recs
( F )  <->  <. x ,  u >.  e.  U. A
)
5 eluni 3831 . . . . . . . 8  |-  ( <.
x ,  u >.  e. 
U. A  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
64, 5bitri 242 . . . . . . 7  |-  ( <.
x ,  u >.  e. recs
( F )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
73eleq2i 2348 . . . . . . . 8  |-  ( <.
x ,  v >.  e. recs ( F )  <->  <. x ,  v >.  e.  U. A
)
8 eluni 3831 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  U. A  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
97, 8bitri 242 . . . . . . 7  |-  ( <.
x ,  v >.  e. recs ( F )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
106, 9anbi12i 680 . . . . . 6  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  <-> 
( E. g (
<. x ,  u >.  e.  g  /\  g  e.  A )  /\  E. h ( <. x ,  v >.  e.  h  /\  h  e.  A
) ) )
11 eeanv 1855 . . . . . 6  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( E. g ( <. x ,  u >.  e.  g  /\  g  e.  A
)  /\  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) ) )
1210, 11bitr4i 245 . . . . 5  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  <->  E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) ) )
13 an4 799 . . . . . . . 8  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) ) )
14 ancom 439 . . . . . . . 8  |-  ( ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) )  <->  ( (
g  e.  A  /\  h  e.  A )  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) ) )
1513, 14bitri 242 . . . . . . 7  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( (
g  e.  A  /\  h  e.  A )  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) ) )
161tfrlem5 6391 . . . . . . . 8  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( <. x ,  u >.  e.  g  /\  <. x ,  v
>.  e.  h )  ->  u  =  v )
)
1716imp 420 . . . . . . 7  |-  ( ( ( g  e.  A  /\  h  e.  A
)  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v
>.  e.  h ) )  ->  u  =  v )
1815, 17sylbi 189 . . . . . 6  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
1918exlimivv 1668 . . . . 5  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2012, 19sylbi 189 . . . 4  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2120ax-gen 1534 . . 3  |-  A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2221gen2 1535 . 2  |-  A. x A. u A. v ( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v
>.  e. recs ( F ) )  ->  u  =  v )
23 dffun4 5233 . 2  |-  ( Fun recs
( F )  <->  ( Rel recs ( F )  /\  A. x A. u A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
) )
242, 22, 23mpbir2an 888 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   E.wrex 2545   <.cop 3644   U.cuni 3828   Oncon0 4391    |` cres 4690   Rel wrel 4693   Fun wfun 5215    Fn wfn 5216   ` cfv 5221  recscrecs 6382
This theorem is referenced by:  tfrlem9  6396  tfrlem9a  6397  tfrlem10  6398  tfrlem14  6402  tfrlem16  6404  tfr1a  6405  tfr1  6408
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229  df-recs 6383
  Copyright terms: Public domain W3C validator