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Theorem tfrlem3a 7038
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 5667 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( f  Fn  x  <->  G  Fn  z ) )
3 simpll 753 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  f  =  G )
4 simpr 461 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
53, 4fveq12d 5865 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( G `  w ) )
63, 4reseq12d 5267 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( G  |`  w
) )
76fveq2d 5863 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( G  |`  w ) ) )
85, 7eqeq12d 2484 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
9 simpr 461 . . . . . 6  |-  ( ( f  =  G  /\  x  =  z )  ->  x  =  z )
109adantr 465 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
118, 10cbvraldva2 3087 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  <->  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
122, 11anbi12d 710 . . 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 ) ) ) ) )
1312cbvrexdva 3090 . 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 ) ) ) ) )
14 tfrlem3.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
151, 13, 14elab2 3248 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 369    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   _Vcvv 3108   Oncon0 4873    |` cres 4996    Fn wfn 5576   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589
This theorem is referenced by:  tfrlem3  7039  tfrlem5  7041  tfrlem9a  7047
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