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Theorem wfrlem1 25470
Description: Lemma for well-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 Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem1.1  |-  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 )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
Assertion
Ref Expression
wfrlem1  |-  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 )  =  ( 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 wfrlem1
StepHypRef Expression
1 wfrlem1.1 . 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 )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
2 fneq1 5493 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  x  <->  g  Fn  x ) )
3 fveq1 5686 . . . . . . . 8  |-  ( f  =  g  ->  (
f `  y )  =  ( g `  y ) )
4 reseq1 5099 . . . . . . . . 9  |-  ( f  =  g  ->  (
f  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  y )
) )
54fveq2d 5691 . . . . . . . 8  |-  ( f  =  g  ->  ( G `  ( f  |` 
Pred ( R ,  A ,  y )
) )  =  ( G `  ( g  |`  Pred ( R ,  A ,  y )
) ) )
63, 5eqeq12d 2418 . . . . . . 7  |-  ( f  =  g  ->  (
( f `  y
)  =  ( G `
 ( f  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  y )  =  ( G `  ( g  |`  Pred ( R ,  A ,  y )
) ) ) )
76ralbidv 2686 . . . . . 6  |-  ( f  =  g  ->  ( A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  Pred ( R ,  A ,  y ) ) )  <->  A. y  e.  x  ( g `  y
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  y ) ) ) ) )
82, 73anbi13d 1256 . . . . 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  y )
) ) ) ) )
98exbidv 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A , 
y ) ) ) ) ) )
10 fneq2 5494 . . . . . 6  |-  ( x  =  z  ->  (
g  Fn  x  <->  g  Fn  z ) )
11 sseq1 3329 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  A  <->  z  C_  A ) )
12 sseq2 3330 . . . . . . . . 9  |-  ( x  =  z  ->  ( Pred ( R ,  A ,  y )  C_  x 
<-> 
Pred ( R ,  A ,  y )  C_  z ) )
1312raleqbi1dv 2872 . . . . . . . 8  |-  ( x  =  z  ->  ( A. y  e.  x  Pred ( R ,  A ,  y )  C_  x 
<-> 
A. y  e.  z 
Pred ( R ,  A ,  y )  C_  z ) )
14 predeq3 25385 . . . . . . . . . 10  |-  ( y  =  w  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  w ) )
1514sseq1d 3335 . . . . . . . . 9  |-  ( y  =  w  ->  ( Pred ( R ,  A ,  y )  C_  z 
<-> 
Pred ( R ,  A ,  w )  C_  z ) )
1615cbvralv 2892 . . . . . . . 8  |-  ( A. y  e.  z  Pred ( R ,  A , 
y )  C_  z  <->  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)
1713, 16syl6bb 253 . . . . . . 7  |-  ( x  =  z  ->  ( A. y  e.  x  Pred ( R ,  A ,  y )  C_  x 
<-> 
A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z ) )
1811, 17anbi12d 692 . . . . . 6  |-  ( x  =  z  ->  (
( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x )  <->  ( z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w
)  C_  z )
) )
19 raleq 2864 . . . . . . 7  |-  ( x  =  z  ->  ( A. y  e.  x  ( g `  y
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  A. y  e.  z  ( g `  y
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  y ) ) ) ) )
20 fveq2 5687 . . . . . . . . 9  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
2114reseq2d 5105 . . . . . . . . . 10  |-  ( y  =  w  ->  (
g  |`  Pred ( R ,  A ,  y )
)  =  ( g  |`  Pred ( R ,  A ,  w )
) )
2221fveq2d 5691 . . . . . . . . 9  |-  ( y  =  w  ->  ( G `  ( g  |` 
Pred ( R ,  A ,  y )
) )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) )
2320, 22eqeq12d 2418 . . . . . . . 8  |-  ( y  =  w  ->  (
( g `  y
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  ( g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) ) )
2423cbvralv 2892 . . . . . . 7  |-  ( A. y  e.  z  (
g `  y )  =  ( G `  ( g  |`  Pred ( R ,  A , 
y ) ) )  <->  A. w  e.  z 
( g `  w
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  w ) ) ) )
2519, 24syl6bb 253 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  x  ( g `  y
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  y ) ) )  <->  A. w  e.  z  ( g `  w
)  =  ( G `
 ( g  |`  Pred ( R ,  A ,  w ) ) ) ) )
2610, 18, 253anbi123d 1254 . . . . 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) ) ) )
2726cbvexv 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w ) ) ) ) )
289, 27syl6bb 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) )
2928cbvabv 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 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) ) }
301, 29eqtri 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 )  =  ( 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   Predcpred 25381
This theorem is referenced by:  wfrlem2  25471  wfrlem3  25472  wfrlem4  25473  wfrlem9  25478
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-pred 25382
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