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Theorem tfrlem3a 6576
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 5475 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5668 . . . . . . 7  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 5081 . . . . . . . 8  |-  ( f  =  g  ->  (
f  |`  y )  =  ( g  |`  y
) )
54fveq2d 5673 . . . . . . 7  |-  ( f  =  g  ->  ( F `  ( f  |`  y ) )  =  ( F `  (
g  |`  y ) ) )
63, 5eqeq12d 2402 . . . . . 6  |-  ( f  =  g  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( g `  y )  =  ( F `  ( g  |`  y ) ) ) )
76ralbidv 2670 . . . . 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 2671 . . 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 2507 . 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 2408 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 1649   {cab 2374   A.wral 2650   E.wrex 2651   Oncon0 4523    |` cres 4821    Fn wfn 5390   ` cfv 5395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-res 4831  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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