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Theorem tfrlem3 6258
Description: Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
Hypothesis
Ref Expression
tfrlem3.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
Distinct variable groups:    A, g    f,
g, w, x, y, z, F
Allowed substitution hints:    A( x, y, z, w, f)

Proof of Theorem tfrlem3
StepHypRef Expression
1 tfrlem3.1 . . 3  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 vex 2715 . . 3  |-  g  e. 
_V
31, 2tfrlem3a 6257 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
43abbi2i 2272 1  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   {cab 2143   A.wral 2435   E.wrex 2436   Oncon0 4323    |` cres 4588    Fn wfn 5165   ` cfv 5170
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 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by:  tfrlem4  6260  tfrlem8  6265  tfrlemi1  6279  tfrexlem  6281  tfri1d  6282  tfrex  6315
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