MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem3a Structured version   Unicode version

Theorem tfrlem3a 7050
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 5630 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( f  Fn  x  <->  G  Fn  z ) )
3 simpll 758 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  f  =  G )
4 simpr 462 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
53, 4fveq12d 5831 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( G `  w ) )
63, 4reseq12d 5068 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( G  |`  w
) )
76fveq2d 5829 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( G  |`  w ) ) )
85, 7eqeq12d 2443 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
9 simplr 760 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
108, 9cbvraldva2 3000 . . . 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 715 . . 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 3003 . 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 3163 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2414   A.wral 2714   E.wrex 2715   _Vcvv 3022    |` cres 4798   Oncon0 5385    Fn wfn 5539   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-res 4808  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552
This theorem is referenced by:  tfrlem3  7051  tfrlem5  7053  tfrlem9a  7059
  Copyright terms: Public domain W3C validator