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Theorem tfrlem4 5962
Description: Lemma for transfinite recursion.  A is the class of all "acceptable" functions, and  F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
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
tfrlem4  |-  ( g  e.  A  ->  Fun  g )
Distinct variable groups:    f, g, x, y, F    A, g
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem4
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem3 5960 . . 3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2190 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fnfun 5027 . . . 4  |-  ( g  Fn  z  ->  Fun  g )
54adantr 270 . . 3  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  Fun  g )
65rexlimivw 2474 . 2  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  Fun  g )
73, 6sylbi 119 1  |-  ( g  e.  A  ->  Fun  g )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   Oncon0 4126    |` cres 4373   Fun wfun 4926    Fn wfn 4927   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940
This theorem is referenced by:  tfrlem6  5965
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