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Theorem frrlem1 24281
Description: Lemma for founded recursion. The final item we are interested in is the union of acceptable functions  B. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypothesis
Ref Expression
frrlem1.1  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
Assertion
Ref Expression
frrlem1  |-  B  =  { g  |  E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) }
Distinct variable groups:    A, f,
g, w, x, y, z    f, G, g, w, x, y, z    R, f, g, w, x, y, z
Allowed substitution hints:    B( x, y, z, w, f, g)

Proof of Theorem frrlem1
StepHypRef Expression
1 frrlem1.1 . 2  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
2 fneq1 5333 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5524 . . . . . . . . 9  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 4949 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  y )
) )
54oveq2d 5874 . . . . . . . . 9  |-  ( f  =  g  ->  (
y G ( f  |`  Pred ( R ,  A ,  y )
) )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) )
63, 5eqeq12d 2297 . . . . . . . 8  |-  ( f  =  g  ->  (
( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) ) )
76ralbidv 2563 . . . . . . 7  |-  ( f  =  g  ->  ( A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) )  <->  A. y  e.  x  ( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) ) ) )
873anbi3d 1258 . . . . . 6  |-  ( f  =  g  ->  (
( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y )
) ) )  <->  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A , 
y ) ) ) ) ) )
92, 8anbi12d 691 . . . . 5  |-  ( f  =  g  ->  (
( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y )
) ) ) )  <-> 
( g  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) ) ) ) )
109exbidv 1612 . . . 4  |-  ( f  =  g  ->  ( E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) )  <->  E. x
( g  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) ) ) ) )
11 fneq2 5334 . . . . . 6  |-  ( x  =  z  ->  (
g  Fn  x  <->  g  Fn  z ) )
12 sseq1 3199 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
13 sseq2 3200 . . . . . . . . 9  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  y )  C_  x 
<-> 
Pred ( R ,  A ,  y )  C_  z ) )
1413raleqbi1dv 2744 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  Pred ( R ,  A ,  y )  C_  x 
<-> 
A. y  e.  z 
Pred ( R ,  A ,  y )  C_  z ) )
15 predeq3 24171 . . . . . . . . . 10  |-  ( y  =  w  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  w ) )
1615sseq1d 3205 . . . . . . . . 9  |-  ( y  =  w  ->  ( Pred ( R ,  A ,  y )  C_  z 
<-> 
Pred ( R ,  A ,  w )  C_  z ) )
1716cbvralv 2764 . . . . . . . 8  |-  ( A. y  e.  z  Pred ( R ,  A , 
y )  C_  z  <->  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)
1814, 17syl6bb 252 . . . . . . 7  |-  ( x  =  z  ->  ( A. y  e.  x  Pred ( R ,  A ,  y )  C_  x 
<-> 
A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z ) )
19 raleq 2736 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  ( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  A. y  e.  z  ( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) ) ) )
20 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
21 id 19 . . . . . . . . . . 11  |-  ( y  =  w  ->  y  =  w )
2215reseq2d 4955 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
g  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  w )
) )
2321, 22oveq12d 5876 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y G ( g  |`  Pred ( R ,  A ,  y )
) )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) )
2420, 23eqeq12d 2297 . . . . . . . . 9  |-  ( y  =  w  ->  (
( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) )
2524cbvralv 2764 . . . . . . . 8  |-  ( A. y  e.  z  (
g `  y )  =  ( y G ( g  |`  Pred ( R ,  A , 
y ) ) )  <->  A. w  e.  z 
( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) )
2619, 25syl6bb 252 . . . . . . 7  |-  ( x  =  z  ->  ( A. y  e.  x  ( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) )
2712, 18, 263anbi123d 1252 . . . . . 6  |-  ( x  =  z  ->  (
( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) )  <->  ( z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w
)  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) )
2811, 27anbi12d 691 . . . . 5  |-  ( x  =  z  ->  (
( g  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) ) )  <-> 
( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) ) ) )
2928cbvexv 1943 . . . 4  |-  ( E. x ( g  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) ) ) )  <->  E. z
( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) ) )
3010, 29syl6bb 252 . . 3  |-  ( f  =  g  ->  ( E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) )  <->  E. z
( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) ) ) )
3130cbvabv 2402 . 2  |-  { f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }  =  { g  |  E. z ( g  Fn  z  /\  (
z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) }
321, 31eqtri 2303 1  |-  B  =  { g  |  E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623   {cab 2269   A.wral 2543    C_ wss 3152    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Predcpred 24167
This theorem is referenced by:  frrlem2  24282  frrlem3  24283  frrlem4  24284  frrlem5e  24289
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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-pred 24168
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