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Theorem tz7.44-2 6374
Description: The value of  F at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1  |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `
 U. dom  x
) ) ) ) )
tz7.44.2  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
tz7.44.3  |-  ( y  e.  X  ->  ( F  |`  y )  e. 
_V )
tz7.44.4  |-  F  Fn  X
tz7.44.5  |-  Ord  X
Assertion
Ref Expression
tz7.44-2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( H `
 ( F `  B ) ) )
Distinct variable groups:    x, A    x, y, B    x, F, y    y, G    x, H    y, X
Allowed substitution hints:    A( y)    G( x)    H( y)    X( x)

Proof of Theorem tz7.44-2
StepHypRef Expression
1 fveq2 5444 . . . 4  |-  ( y  =  suc  B  -> 
( F `  y
)  =  ( F `
 suc  B )
)
2 reseq2 4924 . . . . 5  |-  ( y  =  suc  B  -> 
( F  |`  y
)  =  ( F  |`  suc  B ) )
32fveq2d 5448 . . . 4  |-  ( y  =  suc  B  -> 
( G `  ( F  |`  y ) )  =  ( G `  ( F  |`  suc  B
) ) )
41, 3eqeq12d 2270 . . 3  |-  ( y  =  suc  B  -> 
( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  suc  B )  =  ( G `  ( F  |`  suc  B ) ) ) )
5 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
64, 5vtoclga 2817 . 2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( G `
 ( F  |`  suc  B ) ) )
72eleq1d 2322 . . . 4  |-  ( y  =  suc  B  -> 
( ( F  |`  y )  e.  _V  <->  ( F  |`  suc  B )  e.  _V ) )
8 tz7.44.3 . . . 4  |-  ( y  e.  X  ->  ( F  |`  y )  e. 
_V )
97, 8vtoclga 2817 . . 3  |-  ( suc 
B  e.  X  -> 
( F  |`  suc  B
)  e.  _V )
10 noel 3420 . . . . . . 7  |-  -.  B  e.  (/)
11 dmeq 4853 . . . . . . . . 9  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  (  F  |`  suc  B )  =  dom  (/) )
12 dm0 4866 . . . . . . . . 9  |-  dom  (/)  =  (/)
1311, 12syl6eq 2304 . . . . . . . 8  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  (  F  |`  suc  B )  =  (/) )
14 tz7.44.5 . . . . . . . . . . . . 13  |-  Ord  X
15 ordsson 4539 . . . . . . . . . . . . 13  |-  ( Ord 
X  ->  X  C_  On )
1614, 15ax-mp 10 . . . . . . . . . . . 12  |-  X  C_  On
17 ordtr 4364 . . . . . . . . . . . . . 14  |-  ( Ord 
X  ->  Tr  X
)
1814, 17ax-mp 10 . . . . . . . . . . . . 13  |-  Tr  X
19 trsuc 4434 . . . . . . . . . . . . 13  |-  ( ( Tr  X  /\  suc  B  e.  X )  ->  B  e.  X )
2018, 19mpan 654 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  B  e.  X )
2116, 20sseldi 3139 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  ->  B  e.  On )
22 sucidg 4428 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  e.  suc  B )
2321, 22syl 17 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  B  e.  suc  B )
24 dmres 4950 . . . . . . . . . . 11  |-  dom  (  F  |`  suc  B )  =  ( suc  B  i^i  dom  F )
25 ordelss 4366 . . . . . . . . . . . . . 14  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  C_  X )
2614, 25mpan 654 . . . . . . . . . . . . 13  |-  ( suc 
B  e.  X  ->  suc  B  C_  X )
27 tz7.44.4 . . . . . . . . . . . . . 14  |-  F  Fn  X
28 fndm 5267 . . . . . . . . . . . . . 14  |-  ( F  Fn  X  ->  dom  F  =  X )
2927, 28ax-mp 10 . . . . . . . . . . . . 13  |-  dom  F  =  X
3026, 29syl6sseqr 3186 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  suc  B  C_  dom  F )
31 df-ss 3127 . . . . . . . . . . . 12  |-  ( suc 
B  C_  dom  F  <->  ( suc  B  i^i  dom  F )  =  suc  B )
3230, 31sylib 190 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  -> 
( suc  B  i^i  dom 
F )  =  suc  B )
3324, 32syl5eq 2300 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  dom  (  F  |`  suc  B
)  =  suc  B
)
3423, 33eleqtrrd 2333 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  B  e.  dom  (  F  |`  suc  B ) )
35 eleq2 2317 . . . . . . . . 9  |-  ( dom  (  F  |`  suc  B
)  =  (/)  ->  ( B  e.  dom  (  F  |`  suc  B )  <->  B  e.  (/) ) )
3634, 35syl5ibcom 213 . . . . . . . 8  |-  ( suc 
B  e.  X  -> 
( dom  (  F  |` 
suc  B )  =  (/)  ->  B  e.  (/) ) )
3713, 36syl5 30 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B )  =  (/)  ->  B  e.  (/) ) )
3810, 37mtoi 171 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  ( F  |`  suc  B
)  =  (/) )
39 iffalse 3532 . . . . . 6  |-  ( -.  ( F  |`  suc  B
)  =  (/)  ->  if ( ( F  |`  suc  B )  =  (/) ,  A ,  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) ) )  =  if ( Lim 
dom  (  F  |`  suc  B ) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) )
4038, 39syl 17 . . . . 5  |-  ( suc 
B  e.  X  ->  if ( ( F  |`  suc  B )  =  (/) ,  A ,  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) ) )  =  if ( Lim 
dom  (  F  |`  suc  B ) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) )
41 nlimsucg 4591 . . . . . . . 8  |-  ( B  e.  On  ->  -.  Lim  suc  B )
4221, 41syl 17 . . . . . . 7  |-  ( suc 
B  e.  X  ->  -.  Lim  suc  B )
43 limeq 4362 . . . . . . . 8  |-  ( dom  (  F  |`  suc  B
)  =  suc  B  ->  ( Lim  dom  (  F  |`  suc  B )  <->  Lim  suc  B ) )
4433, 43syl 17 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( Lim  dom  (  F  |`  suc  B )  <->  Lim  suc  B
) )
4542, 44mtbird 294 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  Lim  dom  (  F  |` 
suc  B ) )
46 iffalse 3532 . . . . . 6  |-  ( -. 
Lim  dom  (  F  |`  suc  B )  ->  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) )  =  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) )
4745, 46syl 17 . . . . 5  |-  ( suc 
B  e.  X  ->  if ( Lim  dom  (  F  |`  suc  B ) ,  U. ran  (  F  |`  suc  B ) ,  ( H `  ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) ) ) )  =  ( H `  ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) ) ) )
4833unieqd 3798 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. dom  (  F  |`  suc  B )  =  U. suc  B )
49 eloni 4360 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
50 ordunisuc 4581 . . . . . . . . . . 11  |-  ( Ord 
B  ->  U. suc  B  =  B )
5149, 50syl 17 . . . . . . . . . 10  |-  ( B  e.  On  ->  U. suc  B  =  B )
5221, 51syl 17 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. suc  B  =  B )
5348, 52eqtrd 2288 . . . . . . . 8  |-  ( suc 
B  e.  X  ->  U. dom  (  F  |`  suc  B )  =  B )
5453fveq2d 5448 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) )  =  ( ( F  |`  suc  B
) `  B )
)
55 fvres 5461 . . . . . . . 8  |-  ( B  e.  suc  B  -> 
( ( F  |`  suc  B ) `  B
)  =  ( F `
 B ) )
5623, 55syl 17 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  B
)  =  ( F `
 B ) )
5754, 56eqtrd 2288 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) )  =  ( F `  B ) )
5857fveq2d 5448 . . . . 5  |-  ( suc 
B  e.  X  -> 
( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) )  =  ( H `  ( F `
 B ) ) )
5940, 47, 583eqtrd 2292 . . . 4  |-  ( suc 
B  e.  X  ->  if ( ( F  |`  suc  B )  =  (/) ,  A ,  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) ) )  =  ( H `  ( F `  B ) ) )
60 fvex 5458 . . . 4  |-  ( H `
 ( F `  B ) )  e. 
_V
6159, 60syl6eqel 2344 . . 3  |-  ( suc 
B  e.  X  ->  if ( ( F  |`  suc  B )  =  (/) ,  A ,  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) ) )  e.  _V )
62 eqeq1 2262 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  (
x  =  (/)  <->  ( F  |` 
suc  B )  =  (/) ) )
63 dmeq 4853 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  dom  x  =  dom  (  F  |`  suc  B ) )
64 limeq 4362 . . . . . . 7  |-  ( dom  x  =  dom  (  F  |`  suc  B )  ->  ( Lim  dom  x 
<->  Lim  dom  (  F  |` 
suc  B ) ) )
6563, 64syl 17 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( Lim  dom  x  <->  Lim  dom  (  F  |`  suc  B ) ) )
66 rneq 4878 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  ran  x  =  ran  (  F  |`  suc  B ) )
6766unieqd 3798 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  U. ran  x  =  U. ran  (  F  |`  suc  B ) )
68 fveq1 5443 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  x
) )
6963unieqd 3798 . . . . . . . . 9  |-  ( x  =  ( F  |`  suc  B )  ->  U. dom  x  =  U. dom  (  F  |`  suc  B ) )
7069fveq2d 5448 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
( F  |`  suc  B
) `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  (  F  |`  suc  B ) ) )
7168, 70eqtrd 2288 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  (  F  |`  suc  B ) ) )
7271fveq2d 5448 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( H `  ( x `  U. dom  x ) )  =  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) )
7365, 67, 72ifbieq12d 3547 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `
 U. dom  x
) ) )  =  if ( Lim  dom  (  F  |`  suc  B
) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) )
7462, 73ifbieq2d 3545 . . . 4  |-  ( x  =  ( F  |`  suc  B )  ->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `  U. dom  x ) ) ) )  =  if ( ( F  |`  suc  B
)  =  (/) ,  A ,  if ( Lim  dom  (  F  |`  suc  B
) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) ) )
75 tz7.44.1 . . . 4  |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `
 U. dom  x
) ) ) ) )
7674, 75fvmptg 5520 . . 3  |-  ( ( ( F  |`  suc  B
)  e.  _V  /\  if ( ( F  |`  suc  B )  =  (/) ,  A ,  if ( Lim  dom  (  F  |` 
suc  B ) , 
U. ran  (  F  |` 
suc  B ) ,  ( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) ) ) )  e.  _V )  -> 
( G `  ( F  |`  suc  B ) )  =  if ( ( F  |`  suc  B
)  =  (/) ,  A ,  if ( Lim  dom  (  F  |`  suc  B
) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) ) )
779, 61, 76syl2anc 645 . 2  |-  ( suc 
B  e.  X  -> 
( G `  ( F  |`  suc  B ) )  =  if ( ( F  |`  suc  B
)  =  (/) ,  A ,  if ( Lim  dom  (  F  |`  suc  B
) ,  U. ran  (  F  |`  suc  B
) ,  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) ) ) )
786, 77, 593eqtrd 2292 1  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( H `
 ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   _Vcvv 2757    i^i cin 3112    C_ wss 3113   (/)c0 3416   ifcif 3525   U.cuni 3787    e. cmpt 4037   Tr wtr 4073   Ord word 4349   Oncon0 4350   Lim wlim 4351   suc csuc 4352   dom cdm 4647   ran crn 4648    |` cres 4649    Fn wfn 4654   ` cfv 4659
This theorem is referenced by:  rdgsucg  6390
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-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-fv 4675
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