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Theorem tfrlem4 6482
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 variable  z is 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 6480 . . 3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
32abeq2i 2465 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
4 fnfun 5423 . . . 4  |-  ( g  Fn  z  ->  Fun  g )
54adantr 451 . . 3  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  Fun  g )
65rexlimivw 2739 . 2  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  Fun  g )
73, 6sylbi 187 1  |-  ( g  e.  A  ->  Fun  g )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   E.wrex 2620   Oncon0 4474    |` cres 4773   Fun wfun 5331    Fn wfn 5332   ` cfv 5337
This theorem is referenced by:  tfrlem6  6485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-res 4783  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
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