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Theorem abianfplem 6470
Description: Lemma for abianfp 6471. We prove by transfinite induction that if  F has a fixed point  x, then its iterates also equal  x. This lemma is used for the "trivial" direction of the main theorem. (Contributed by NM, 4-Sep-2004.)
Hypotheses
Ref Expression
abianfp.1  |-  A  e. 
_V
abianfp.2  |-  G  =  rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x )
Assertion
Ref Expression
abianfplem  |-  ( v  e.  On  ->  (
( F `  x
)  =  x  -> 
( G `  v
)  =  x ) )
Distinct variable groups:    x, v    z, v, F    v, G
Allowed substitution hints:    A( x, z, v)    F( x)    G( x, z)

Proof of Theorem abianfplem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( v  =  (/)  ->  ( G `
 v )  =  ( G `  (/) ) )
21eqeq1d 2291 . 2  |-  ( v  =  (/)  ->  ( ( G `  v )  =  x  <->  ( G `  (/) )  =  x ) )
3 fveq2 5525 . . 3  |-  ( v  =  y  ->  ( G `  v )  =  ( G `  y ) )
43eqeq1d 2291 . 2  |-  ( v  =  y  ->  (
( G `  v
)  =  x  <->  ( G `  y )  =  x ) )
5 fveq2 5525 . . 3  |-  ( v  =  suc  y  -> 
( G `  v
)  =  ( G `
 suc  y )
)
65eqeq1d 2291 . 2  |-  ( v  =  suc  y  -> 
( ( G `  v )  =  x  <-> 
( G `  suc  y )  =  x ) )
7 abianfp.2 . . . . 5  |-  G  =  rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x )
87fveq1i 5526 . . . 4  |-  ( G `
 (/) )  =  ( rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x ) `  (/) )
9 vex 2791 . . . . 5  |-  x  e. 
_V
109rdg0 6434 . . . 4  |-  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  (/) )  =  x
118, 10eqtri 2303 . . 3  |-  ( G `
 (/) )  =  x
1211a1i 10 . 2  |-  ( ( F `  x )  =  x  ->  ( G `  (/) )  =  x )
13 fvex 5539 . . . . 5  |-  ( F `
 ( G `  y ) )  e. 
_V
14 fveq2 5525 . . . . . 6  |-  ( v  =  z  ->  ( F `  v )  =  ( F `  z ) )
15 fveq2 5525 . . . . . 6  |-  ( v  =  ( G `  y )  ->  ( F `  v )  =  ( F `  ( G `  y ) ) )
167, 14, 15rdgsucmpt2 6443 . . . . 5  |-  ( ( y  e.  On  /\  ( F `  ( G `
 y ) )  e.  _V )  -> 
( G `  suc  y )  =  ( F `  ( G `
 y ) ) )
1713, 16mpan2 652 . . . 4  |-  ( y  e.  On  ->  ( G `  suc  y )  =  ( F `  ( G `  y ) ) )
18 fveq2 5525 . . . . 5  |-  ( ( G `  y )  =  x  ->  ( F `  ( G `  y ) )  =  ( F `  x
) )
19 id 19 . . . . 5  |-  ( ( F `  x )  =  x  ->  ( F `  x )  =  x )
2018, 19sylan9eqr 2337 . . . 4  |-  ( ( ( F `  x
)  =  x  /\  ( G `  y )  =  x )  -> 
( F `  ( G `  y )
)  =  x )
2117, 20sylan9eq 2335 . . 3  |-  ( ( y  e.  On  /\  ( ( F `  x )  =  x  /\  ( G `  y )  =  x ) )  ->  ( G `  suc  y )  =  x )
2221exp32 588 . 2  |-  ( y  e.  On  ->  (
( F `  x
)  =  x  -> 
( ( G `  y )  =  x  ->  ( G `  suc  y )  =  x ) ) )
23 vex 2791 . . . . . . . 8  |-  v  e. 
_V
24 rdglim2a 6446 . . . . . . . 8  |-  ( ( v  e.  _V  /\  Lim  v )  ->  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  v
)  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
) )
2523, 24mpan 651 . . . . . . 7  |-  ( Lim  v  ->  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  v
)  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
) )
267fveq1i 5526 . . . . . . 7  |-  ( G `
 v )  =  ( rec ( ( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 v )
277fveq1i 5526 . . . . . . . . 9  |-  ( G `
 y )  =  ( rec ( ( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 y )
2827a1i 10 . . . . . . . 8  |-  ( y  e.  v  ->  ( G `  y )  =  ( rec (
( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 y ) )
2928iuneq2i 3923 . . . . . . 7  |-  U_ y  e.  v  ( G `  y )  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
)
3025, 26, 293eqtr4g 2340 . . . . . 6  |-  ( Lim  v  ->  ( G `  v )  =  U_ y  e.  v  ( G `  y )
)
3130adantr 451 . . . . 5  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  ( G `  v )  =  U_ y  e.  v  ( G `  y
) )
32 iuneq2 3921 . . . . . 6  |-  ( A. y  e.  v  ( G `  y )  =  x  ->  U_ y  e.  v  ( G `  y )  =  U_ y  e.  v  x
)
33 df-lim 4397 . . . . . . . 8  |-  ( Lim  v  <->  ( Ord  v  /\  v  =/=  (/)  /\  v  =  U. v ) )
3433simp2bi 971 . . . . . . 7  |-  ( Lim  v  ->  v  =/=  (/) )
35 iunconst 3913 . . . . . . 7  |-  ( v  =/=  (/)  ->  U_ y  e.  v  x  =  x )
3634, 35syl 15 . . . . . 6  |-  ( Lim  v  ->  U_ y  e.  v  x  =  x )
3732, 36sylan9eqr 2337 . . . . 5  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  U_ y  e.  v  ( G `  y )  =  x )
3831, 37eqtrd 2315 . . . 4  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  ( G `  v )  =  x )
3938ex 423 . . 3  |-  ( Lim  v  ->  ( A. y  e.  v  ( G `  y )  =  x  ->  ( G `
 v )  =  x ) )
4039a1d 22 . 2  |-  ( Lim  v  ->  ( ( F `  x )  =  x  ->  ( A. y  e.  v  ( G `  y )  =  x  ->  ( G `
 v )  =  x ) ) )
412, 4, 6, 12, 22, 40tfinds2 4654 1  |-  ( v  e.  On  ->  (
( F `  x
)  =  x  -> 
( G `  v
)  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   (/)c0 3455   U.cuni 3827   U_ciun 3905    e. cmpt 4077   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  abianfp  6471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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