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Theorem tfrlem3a 6631
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 5526 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5719 . . . . . . 7  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 5132 . . . . . . . 8  |-  ( f  =  g  ->  (
f  |`  y )  =  ( g  |`  y
) )
54fveq2d 5724 . . . . . . 7  |-  ( f  =  g  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
g  |`  y ) ) )
63, 5eqeq12d 2449 . . . . . 6  |-  ( f  =  g  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
76ralbidv 2717 . . . . 5  |-  ( f  =  g  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  A. y  e.  x  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
82, 7anbi12d 692 . . . 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 2718 . . 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 2554 . 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 2455 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 359    = wceq 1652   {cab 2421   A.wral 2697   E.wrex 2698   Oncon0 4573    |` cres 4872    Fn wfn 5441   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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