ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem7 Unicode version

Theorem tfrlem7 6180
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
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
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem7
Dummy variables  g  h  u  v are mutually distinct and distinct from all other variables.
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 6179 . 2  |-  Rel recs ( F )
31recsfval 6178 . . . . . . . . 9  |- recs ( F )  =  U. A
43eleq2i 2182 . . . . . . . 8  |-  ( <.
x ,  u >.  e. recs
( F )  <->  <. x ,  u >.  e.  U. A
)
5 eluni 3707 . . . . . . . 8  |-  ( <.
x ,  u >.  e. 
U. A  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
64, 5bitri 183 . . . . . . 7  |-  ( <.
x ,  u >.  e. recs
( F )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
73eleq2i 2182 . . . . . . . 8  |-  ( <.
x ,  v >.  e. recs ( F )  <->  <. x ,  v >.  e.  U. A
)
8 eluni 3707 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  U. A  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
97, 8bitri 183 . . . . . . 7  |-  ( <.
x ,  v >.  e. recs ( F )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
106, 9anbi12i 453 . . . . . 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 1882 . . . . . 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 186 . . . . 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 df-br 3898 . . . . . . . . 9  |-  ( x g u  <->  <. x ,  u >.  e.  g
)
14 df-br 3898 . . . . . . . . 9  |-  ( x h v  <->  <. x ,  v >.  e.  h
)
1513, 14anbi12i 453 . . . . . . . 8  |-  ( ( x g u  /\  x h v )  <-> 
( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) )
161tfrlem5 6177 . . . . . . . . 9  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
1716impcom 124 . . . . . . . 8  |-  ( ( ( x g u  /\  x h v )  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1815, 17sylanbr 281 . . . . . . 7  |-  ( ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1918an4s 560 . . . . . 6  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2019exlimivv 1850 . . . . 5  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2112, 20sylbi 120 . . . 4  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2221ax-gen 1408 . . 3  |-  A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2322gen2 1409 . 2  |-  A. x A. u A. v ( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v
>.  e. recs ( F ) )  ->  u  =  v )
24 dffun4 5102 . 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 )
) )
252, 23, 24mpbir2an 909 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   <.cop 3498   U.cuni 3704   class class class wbr 3897   Oncon0 4253    |` cres 4509   Rel wrel 4512   Fun wfun 5085    Fn wfn 5086   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrlem9  6182  tfrfun  6183  tfrlemibfn  6191  tfrlemiubacc  6193  tfri1d  6198  rdgfun  6236
  Copyright terms: Public domain W3C validator