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Theorem tfrlem7 6317
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 6316 . 2  |-  Rel recs ( F )
31recsfval 6315 . . . . . . . . 9  |- recs ( F )  =  U. A
43eleq2i 2244 . . . . . . . 8  |-  ( <.
x ,  u >.  e. recs
( F )  <->  <. x ,  u >.  e.  U. A
)
5 eluni 3812 . . . . . . . 8  |-  ( <.
x ,  u >.  e. 
U. A  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
64, 5bitri 184 . . . . . . 7  |-  ( <.
x ,  u >.  e. recs
( F )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
73eleq2i 2244 . . . . . . . 8  |-  ( <.
x ,  v >.  e. recs ( F )  <->  <. x ,  v >.  e.  U. A
)
8 eluni 3812 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  U. A  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
97, 8bitri 184 . . . . . . 7  |-  ( <.
x ,  v >.  e. recs ( F )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
106, 9anbi12i 460 . . . . . 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 1932 . . . . . 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 187 . . . . 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 4004 . . . . . . . . 9  |-  ( x g u  <->  <. x ,  u >.  e.  g
)
14 df-br 4004 . . . . . . . . 9  |-  ( x h v  <->  <. x ,  v >.  e.  h
)
1513, 14anbi12i 460 . . . . . . . 8  |-  ( ( x g u  /\  x h v )  <-> 
( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) )
161tfrlem5 6314 . . . . . . . . 9  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
1716impcom 125 . . . . . . . 8  |-  ( ( ( x g u  /\  x h v )  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1815, 17sylanbr 285 . . . . . . 7  |-  ( ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1918an4s 588 . . . . . 6  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2019exlimivv 1896 . . . . 5  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2112, 20sylbi 121 . . . 4  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2221ax-gen 1449 . . 3  |-  A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2322gen2 1450 . 2  |-  A. x A. u A. v ( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v
>.  e. recs ( F ) )  ->  u  =  v )
24 dffun4 5227 . 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 942 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3595   U.cuni 3809   class class class wbr 4003   Oncon0 4363    |` cres 4628   Rel wrel 4631   Fun wfun 5210    Fn wfn 5211   ` cfv 5216  recscrecs 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-recs 6305
This theorem is referenced by:  tfrlem9  6319  tfrfun  6320  tfrlemibfn  6328  tfrlemiubacc  6330  tfri1d  6335  rdgfun  6373
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