ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem4 Unicode version

Theorem tfrlem4 6292
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 6290 . . 3  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) ) }
32abeq2i 2281 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )
4 fnfun 5295 . . . 4  |-  ( g  Fn  z  ->  Fun  g )
54adantr 274 . . 3  |-  ( ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w
) ) )  ->  Fun  g )
65rexlimivw 2583 . 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 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   Oncon0 4348    |` cres 4613   Fun wfun 5192    Fn wfn 5193   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by:  tfrlem6  6295
  Copyright terms: Public domain W3C validator