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Theorem tfrlem4 6607
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 6605 . . 3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
32abeq2i 2519 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
4 fnfun 5509 . . . 4  |-  ( g  Fn  z  ->  Fun  g )
54adantr 452 . . 3  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  Fun  g )
65rexlimivw 2794 . 2  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  ->  Fun  g )
73, 6sylbi 188 1  |-  ( g  e.  A  ->  Fun  g )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2398   A.wral 2674   E.wrex 2675   Oncon0 4549    |` cres 4847   Fun wfun 5415    Fn wfn 5416   ` cfv 5421
This theorem is referenced by:  tfrlem6  6610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429
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