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Theorem tfrlem4 6262
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 6260 . . 3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2268 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fnfun 5269 . . . 4  |-  ( g  Fn  z  ->  Fun  g )
54adantr 274 . . 3  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  Fun  g )
65rexlimivw 2570 . 2  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  Fun  g )
73, 6sylbi 120 1  |-  ( g  e.  A  ->  Fun  g )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   Oncon0 4325    |` cres 4590   Fun wfun 5166    Fn wfn 5167   ` cfv 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180
This theorem is referenced by:  tfrlem6  6265
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