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Theorem tfrlem3a 6410
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, 22-Jul-2012.)
Hypothesis
Ref Expression
tfrlem3a.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem3a  |-  A  =  { g  |  E. x  e.  On  (
g  Fn  x  /\  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
Distinct variable groups:    x, y,
f, g    f, F, g
Allowed substitution hints:    A( x, y, f, g)    F( x, y)

Proof of Theorem tfrlem3a
StepHypRef Expression
1 tfrlem3a.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 fneq1 5349 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5540 . . . . . . 7  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 4965 . . . . . . . 8  |-  ( f  =  g  ->  (
f  |`  y )  =  ( g  |`  y
) )
54fveq2d 5545 . . . . . . 7  |-  ( f  =  g  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
g  |`  y ) ) )
63, 5eqeq12d 2310 . . . . . 6  |-  ( f  =  g  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
76ralbidv 2576 . . . . 5  |-  ( f  =  g  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
82, 7anbi12d 691 . . . 4  |-  ( 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 ) ) ) ) )
98rexbidv 2577 . . 3  |-  ( 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 ) ) ) ) )
109cbvabv 2415 . 2  |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }  =  { g  |  E. x  e.  On  ( g  Fn  x  /\  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) }
111, 10eqtri 2316 1  |-  A  =  { g  |  E. x  e.  On  (
g  Fn  x  /\  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y
) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   {cab 2282   A.wral 2556   E.wrex 2557   Oncon0 4408    |` cres 4707    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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