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Theorem tfrlem7 7044
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 7043 . 2  |-  Rel recs ( F )
31recsfval 7042 . . . . . . . . 9  |- recs ( F )  =  U. A
43eleq2i 2532 . . . . . . . 8  |-  ( <.
x ,  u >.  e. recs
( F )  <->  <. x ,  u >.  e.  U. A
)
5 eluni 4238 . . . . . . . 8  |-  ( <.
x ,  u >.  e. 
U. A  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
64, 5bitri 249 . . . . . . 7  |-  ( <.
x ,  u >.  e. recs
( F )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
73eleq2i 2532 . . . . . . . 8  |-  ( <.
x ,  v >.  e. recs ( F )  <->  <. x ,  v >.  e.  U. A
)
8 eluni 4238 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  U. A  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
97, 8bitri 249 . . . . . . 7  |-  ( <.
x ,  v >.  e. recs ( F )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
106, 9anbi12i 695 . . . . . 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 1993 . . . . . 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 252 . . . . 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 4440 . . . . . . . . 9  |-  ( x g u  <->  <. x ,  u >.  e.  g
)
14 df-br 4440 . . . . . . . . 9  |-  ( x h v  <->  <. x ,  v >.  e.  h
)
1513, 14anbi12i 695 . . . . . . . 8  |-  ( ( x g u  /\  x h v )  <-> 
( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) )
161tfrlem5 7041 . . . . . . . . 9  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
1716impcom 428 . . . . . . . 8  |-  ( ( ( x g u  /\  x h v )  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1815, 17sylanbr 471 . . . . . . 7  |-  ( ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) )  ->  u  =  v )
1918an4s 824 . . . . . 6  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2019exlimivv 1728 . . . . 5  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2112, 20sylbi 195 . . . 4  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2221ax-gen 1623 . . 3  |-  A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2322gen2 1624 . 2  |-  A. x A. u A. v ( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v
>.  e. recs ( F ) )  ->  u  =  v )
24 dffun4 5582 . 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 918 1  |-  Fun recs ( F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   <.cop 4022   U.cuni 4235   class class class wbr 4439   Oncon0 4867    |` cres 4990   Rel wrel 4993   Fun wfun 5564    Fn wfn 5565   ` cfv 5570  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-recs 7034
This theorem is referenced by:  tfrlem9  7046  tfrlem9a  7047  tfrlem10  7048  tfrlem14  7052  tfrlem16  7054  tfr1a  7055  tfr1  7058
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