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Theorem frrlem1 25495
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 5493 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5686 . . . . . . . . 9  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 5099 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  y )
) )
54oveq2d 6056 . . . . . . . . 9  |-  ( f  =  g  ->  (
y G ( f  |`  Pred ( R ,  A ,  y )
) )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) )
63, 5eqeq12d 2418 . . . . . . . 8  |-  ( f  =  g  ->  (
( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  y )  =  ( y G ( g  |`  Pred ( R ,  A ,  y )
) ) ) )
76ralbidv 2686 . . . . . . 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 1260 . . . . . 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 692 . . . . 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 1633 . . . 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 5494 . . . . . 6  |-  ( x  =  z  ->  (
g  Fn  x  <->  g  Fn  z ) )
12 sseq1 3329 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
13 sseq2 3330 . . . . . . . . 9  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  y )  C_  x 
<-> 
Pred ( R ,  A ,  y )  C_  z ) )
1413raleqbi1dv 2872 . . . . . . . 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 25385 . . . . . . . . . 10  |-  ( y  =  w  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  w ) )
1615sseq1d 3335 . . . . . . . . 9  |-  ( y  =  w  ->  ( Pred ( R ,  A ,  y )  C_  z 
<-> 
Pred ( R ,  A ,  w )  C_  z ) )
1716cbvralv 2892 . . . . . . . 8  |-  ( A. y  e.  z  Pred ( R ,  A , 
y )  C_  z  <->  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)
1814, 17syl6bb 253 . . . . . . 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 2864 . . . . . . . 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 5687 . . . . . . . . . 10  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
21 id 20 . . . . . . . . . . 11  |-  ( y  =  w  ->  y  =  w )
2215reseq2d 5105 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
g  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  w )
) )
2321, 22oveq12d 6058 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y G ( g  |`  Pred ( R ,  A ,  y )
) )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) )
2420, 23eqeq12d 2418 . . . . . . . . 9  |-  ( y  =  w  ->  (
( g `  y
)  =  ( y G ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) )
2524cbvralv 2892 . . . . . . . 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 253 . . . . . . 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 1254 . . . . . 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 692 . . . . 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 2053 . . . 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 253 . . 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 2523 . 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 2424 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 359    /\ w3a 936   E.wex 1547    = wceq 1649   {cab 2390   A.wral 2666    C_ wss 3280    |` cres 4839    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Predcpred 25381
This theorem is referenced by:  frrlem2  25496  frrlem3  25497  frrlem4  25498  frrlem5e  25503
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-ov 6043  df-pred 25382
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