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Theorem abianfplem 6466
Description: Lemma for abianfp 6467. 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
Dummy variable  y is distinct from all other variables.
Allowed substitution hints:    A( x, z, v)    F( x)    G( x, z)

Proof of Theorem abianfplem
StepHypRef Expression
1 fveq2 5486 . . 3  |-  ( v  =  (/)  ->  ( G `
 v )  =  ( G `  (/) ) )
21eqeq1d 2293 . 2  |-  ( v  =  (/)  ->  ( ( G `  v )  =  x  <->  ( G `  (/) )  =  x ) )
3 fveq2 5486 . . 3  |-  ( v  =  y  ->  ( G `  v )  =  ( G `  y ) )
43eqeq1d 2293 . 2  |-  ( v  =  y  ->  (
( G `  v
)  =  x  <->  ( G `  y )  =  x ) )
5 fveq2 5486 . . 3  |-  ( v  =  suc  y  -> 
( G `  v
)  =  ( G `
 suc  y )
)
65eqeq1d 2293 . 2  |-  ( v  =  suc  y  -> 
( ( G `  v )  =  x  <-> 
( G `  suc  y )  =  x ) )
7 abianfp.2 . . . . 5  |-  G  =  rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x )
87fveq1i 5487 . . . 4  |-  ( G `
 (/) )  =  ( rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x ) `  (/) )
9 vex 2793 . . . . 5  |-  x  e. 
_V
109rdg0 6430 . . . 4  |-  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  (/) )  =  x
118, 10eqtri 2305 . . 3  |-  ( G `
 (/) )  =  x
1211a1i 12 . 2  |-  ( ( F `  x )  =  x  ->  ( G `  (/) )  =  x )
13 fvex 5500 . . . . 5  |-  ( F `
 ( G `  y ) )  e. 
_V
14 fveq2 5486 . . . . . 6  |-  ( v  =  z  ->  ( F `  v )  =  ( F `  z ) )
15 fveq2 5486 . . . . . 6  |-  ( v  =  ( G `  y )  ->  ( F `  v )  =  ( F `  ( G `  y ) ) )
167, 14, 15rdgsucmpt2 6439 . . . . 5  |-  ( ( y  e.  On  /\  ( F `  ( G `
 y ) )  e.  _V )  -> 
( G `  suc  y )  =  ( F `  ( G `
 y ) ) )
1713, 16mpan2 654 . . . 4  |-  ( y  e.  On  ->  ( G `  suc  y )  =  ( F `  ( G `  y ) ) )
18 fveq2 5486 . . . . 5  |-  ( ( G `  y )  =  x  ->  ( F `  ( G `  y ) )  =  ( F `  x
) )
19 id 21 . . . . 5  |-  ( ( F `  x )  =  x  ->  ( F `  x )  =  x )
2018, 19sylan9eqr 2339 . . . 4  |-  ( ( ( F `  x
)  =  x  /\  ( G `  y )  =  x )  -> 
( F `  ( G `  y )
)  =  x )
2117, 20sylan9eq 2337 . . 3  |-  ( ( y  e.  On  /\  ( ( F `  x )  =  x  /\  ( G `  y )  =  x ) )  ->  ( G `  suc  y )  =  x )
2221exp32 590 . 2  |-  ( y  e.  On  ->  (
( F `  x
)  =  x  -> 
( ( G `  y )  =  x  ->  ( G `  suc  y )  =  x ) ) )
23 vex 2793 . . . . . . . 8  |-  v  e. 
_V
24 rdglim2a 6442 . . . . . . . 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 653 . . . . . . 7  |-  ( Lim  v  ->  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  v
)  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
) )
267fveq1i 5487 . . . . . . 7  |-  ( G `
 v )  =  ( rec ( ( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 v )
277fveq1i 5487 . . . . . . . . 9  |-  ( G `
 y )  =  ( rec ( ( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 y )
2827a1i 12 . . . . . . . 8  |-  ( y  e.  v  ->  ( G `  y )  =  ( rec (
( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 y ) )
2928iuneq2i 3925 . . . . . . 7  |-  U_ y  e.  v  ( G `  y )  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
)
3025, 26, 293eqtr4g 2342 . . . . . 6  |-  ( Lim  v  ->  ( G `  v )  =  U_ y  e.  v  ( G `  y )
)
3130adantr 453 . . . . 5  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  ( G `  v )  =  U_ y  e.  v  ( G `  y
) )
32 iuneq2 3923 . . . . . 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 973 . . . . . . 7  |-  ( Lim  v  ->  v  =/=  (/) )
35 iunconst 3915 . . . . . . 7  |-  ( v  =/=  (/)  ->  U_ y  e.  v  x  =  x )
3634, 35syl 17 . . . . . 6  |-  ( Lim  v  ->  U_ y  e.  v  x  =  x )
3732, 36sylan9eqr 2339 . . . . 5  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  U_ y  e.  v  ( G `  y )  =  x )
3831, 37eqtrd 2317 . . . 4  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  ( G `  v )  =  x )
3938ex 425 . . 3  |-  ( Lim  v  ->  ( A. y  e.  v  ( G `  y )  =  x  ->  ( G `
 v )  =  x ) )
4039a1d 24 . 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 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448   A.wral 2545   _Vcvv 2790   (/)c0 3457   U.cuni 3829   U_ciun 3907    e. cmpt 4079   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   ` cfv 5222   reccrdg 6418
This theorem is referenced by:  abianfp  6467
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384  df-rdg 6419
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