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Theorem abianfplem 6403
Description: Lemma for abianfp 6404. 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
StepHypRef Expression
1 fveq2 5423 . . 3  |-  ( v  =  (/)  ->  ( G `
 v )  =  ( G `  (/) ) )
21eqeq1d 2264 . 2  |-  ( v  =  (/)  ->  ( ( G `  v )  =  x  <->  ( G `  (/) )  =  x ) )
3 fveq2 5423 . . 3  |-  ( v  =  y  ->  ( G `  v )  =  ( G `  y ) )
43eqeq1d 2264 . 2  |-  ( v  =  y  ->  (
( G `  v
)  =  x  <->  ( G `  y )  =  x ) )
5 fveq2 5423 . . 3  |-  ( v  =  suc  y  -> 
( G `  v
)  =  ( G `
 suc  y )
)
65eqeq1d 2264 . 2  |-  ( v  =  suc  y  -> 
( ( G `  v )  =  x  <-> 
( G `  suc  y )  =  x ) )
7 abianfp.2 . . . . 5  |-  G  =  rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x )
87fveq1i 5424 . . . 4  |-  ( G `
 (/) )  =  ( rec ( ( z  e.  _V  |->  ( F `
 z ) ) ,  x ) `  (/) )
9 vex 2743 . . . . 5  |-  x  e. 
_V
109rdg0 6367 . . . 4  |-  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  (/) )  =  x
118, 10eqtri 2276 . . 3  |-  ( G `
 (/) )  =  x
1211a1i 12 . 2  |-  ( ( F `  x )  =  x  ->  ( G `  (/) )  =  x )
13 fvex 5437 . . . . 5  |-  ( F `
 ( G `  y ) )  e. 
_V
14 fveq2 5423 . . . . . 6  |-  ( v  =  z  ->  ( F `  v )  =  ( F `  z ) )
15 fveq2 5423 . . . . . 6  |-  ( v  =  ( G `  y )  ->  ( F `  v )  =  ( F `  ( G `  y ) ) )
167, 14, 15rdgsucmpt2 6376 . . . . 5  |-  ( ( y  e.  On  /\  ( F `  ( G `
 y ) )  e.  _V )  -> 
( G `  suc  y )  =  ( F `  ( G `
 y ) ) )
1713, 16mpan2 655 . . . 4  |-  ( y  e.  On  ->  ( G `  suc  y )  =  ( F `  ( G `  y ) ) )
18 fveq2 5423 . . . . 5  |-  ( ( G `  y )  =  x  ->  ( F `  ( G `  y ) )  =  ( F `  x
) )
19 id 21 . . . . 5  |-  ( ( F `  x )  =  x  ->  ( F `  x )  =  x )
2018, 19sylan9eqr 2310 . . . 4  |-  ( ( ( F `  x
)  =  x  /\  ( G `  y )  =  x )  -> 
( F `  ( G `  y )
)  =  x )
2117, 20sylan9eq 2308 . . 3  |-  ( ( y  e.  On  /\  ( ( F `  x )  =  x  /\  ( G `  y )  =  x ) )  ->  ( G `  suc  y )  =  x )
2221exp32 591 . 2  |-  ( y  e.  On  ->  (
( F `  x
)  =  x  -> 
( ( G `  y )  =  x  ->  ( G `  suc  y )  =  x ) ) )
23 vex 2743 . . . . . . . 8  |-  v  e. 
_V
24 rdglim2a 6379 . . . . . . . 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 654 . . . . . . 7  |-  ( Lim  v  ->  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  v
)  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
) )
267fveq1i 5424 . . . . . . 7  |-  ( G `
 v )  =  ( rec ( ( z  e.  _V  |->  ( F `  z ) ) ,  x ) `
 v )
277fveq1i 5424 . . . . . . . . 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 3864 . . . . . . 7  |-  U_ y  e.  v  ( G `  y )  =  U_ y  e.  v  ( rec ( ( z  e. 
_V  |->  ( F `  z ) ) ,  x ) `  y
)
3025, 26, 293eqtr4g 2313 . . . . . 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 3862 . . . . . 6  |-  ( A. y  e.  v  ( G `  y )  =  x  ->  U_ y  e.  v  ( G `  y )  =  U_ y  e.  v  x
)
33 df-lim 4334 . . . . . . . 8  |-  ( Lim  v  <->  ( Ord  v  /\  v  =/=  (/)  /\  v  =  U. v ) )
3433simp2bi 976 . . . . . . 7  |-  ( Lim  v  ->  v  =/=  (/) )
35 iunconst 3854 . . . . . . 7  |-  ( v  =/=  (/)  ->  U_ y  e.  v  x  =  x )
3634, 35syl 17 . . . . . 6  |-  ( Lim  v  ->  U_ y  e.  v  x  =  x )
3732, 36sylan9eqr 2310 . . . . 5  |-  ( ( Lim  v  /\  A. y  e.  v  ( G `  y )  =  x )  ->  U_ y  e.  v  ( G `  y )  =  x )
3831, 37eqtrd 2288 . . . 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 4591 1  |-  ( v  e.  On  ->  (
( F `  x
)  =  x  -> 
( G `  v
)  =  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   _Vcvv 2740   (/)c0 3397   U.cuni 3768   U_ciun 3846    e. cmpt 4017   Ord word 4328   Oncon0 4329   Lim wlim 4330   suc csuc 4331   ` cfv 4638   reccrdg 6355
This theorem is referenced by:  abianfp  6404
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356
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