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Theorem tfrlem3 6393
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. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
Distinct variable groups:    x, y,
f, g    x, z,
y, g    f, F, g, x    z, F
Allowed substitution hints:    A( x, y, z, f, g)    F( y)

Proof of Theorem tfrlem3
StepHypRef Expression
1 tfrlem3.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 vex 2791 . . . . 5  |-  g  e. 
_V
3 fneq1 5333 . . . . . . 7  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
4 fveq1 5524 . . . . . . . . 9  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
5 reseq1 4949 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f  |`  y )  =  ( g  |`  y
) )
65fveq2d 5529 . . . . . . . . 9  |-  ( f  =  g  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
g  |`  y ) ) )
74, 6eqeq12d 2297 . . . . . . . 8  |-  ( f  =  g  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
87ralbidv 2563 . . . . . . 7  |-  ( f  =  g  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
93, 8anbi12d 691 . . . . . 6  |-  ( f  =  g  ->  (
( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <-> 
( g  Fn  x  /\  A. y  e.  x  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) ) ) )
109rexbidv 2564 . . . . 5  |-  ( f  =  g  ->  ( E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <->  E. x  e.  On  ( g  Fn  x  /\  A. y  e.  x  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) ) ) )
112, 10elab 2914 . . . 4  |-  ( g  e.  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }  <->  E. x  e.  On  ( g  Fn  x  /\  A. y  e.  x  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) ) )
12 fneq2 5334 . . . . . 6  |-  ( x  =  z  ->  (
g  Fn  x  <->  g  Fn  z ) )
13 raleq 2736 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  x  ( g `  y
)  =  ( F `
 ( g  |`  y ) )  <->  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
1412, 13anbi12d 691 . . . . 5  |-  ( x  =  z  ->  (
( g  Fn  x  /\  A. y  e.  x  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) )  <-> 
( g  Fn  z  /\  A. y  e.  z  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) ) ) )
1514cbvrexv 2765 . . . 4  |-  ( E. x  e.  On  (
g  Fn  x  /\  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y
) ) )  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
1611, 15bitri 240 . . 3  |-  ( g  e.  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }  <->  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y
)  =  ( F `
 ( g  |`  y ) ) ) )
1716abbi2i 2394 . 2  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) }
181, 17eqtri 2303 1  |-  A  =  { g  |  E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   Oncon0 4392    |` cres 4691    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tfrlem4  6395  tfrlem5  6396  tfrlem8  6400  tfrlem9a  6402  rdglem1  6428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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