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Theorem tfrlem3a 7001
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, 9-Apr-1995.)
Hypotheses
Ref Expression
tfrlem3.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlem3.2  |-  G  e. 
_V
Assertion
Ref Expression
tfrlem3a  |-  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
Distinct variable groups:    w, f, x, y, z, F    f, G, w, x, y, z
Allowed substitution hints:    A( x, y, z, w, f)

Proof of Theorem tfrlem3a
StepHypRef Expression
1 tfrlem3.2 . 2  |-  G  e. 
_V
2 fneq12 5609 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( f  Fn  x  <->  G  Fn  z ) )
3 simpll 752 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  f  =  G )
4 simpr 459 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
53, 4fveq12d 5809 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( G `  w ) )
63, 4reseq12d 5214 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( G  |`  w
) )
76fveq2d 5807 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( G  |`  w ) ) )
85, 7eqeq12d 2422 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
9 simplr 754 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
108, 9cbvraldva2 3035 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  <->  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
112, 10anbi12d 709 . . 3  |-  ( ( f  =  G  /\  x  =  z )  ->  ( ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )  <-> 
( G  Fn  z  /\  A. w  e.  z  ( G `  w
)  =  ( F `
 ( G  |`  w ) ) ) ) )
1211cbvrexdva 3038 . 2  |-  ( f  =  G  ->  ( E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w
)  =  ( F `
 ( G  |`  w ) ) ) ) )
13 tfrlem3.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
141, 12, 13elab2 3196 1  |-  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   {cab 2385   A.wral 2751   E.wrex 2752   _Vcvv 3056   Oncon0 4819    |` cres 4942    Fn wfn 5518   ` cfv 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531
This theorem is referenced by:  tfrlem3  7002  tfrlem5  7004  tfrlem9a  7010
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