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Theorem tz7.44-2 6694
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 5757 . . . 4  |-  ( y  =  suc  B  -> 
( F `  y
)  =  ( F `
 suc  B )
)
2 reseq2 5170 . . . . 5  |-  ( y  =  suc  B  -> 
( F  |`  y
)  =  ( F  |`  suc  B ) )
32fveq2d 5761 . . . 4  |-  ( y  =  suc  B  -> 
( G `  ( F  |`  y ) )  =  ( G `  ( F  |`  suc  B
) ) )
41, 3eqeq12d 2456 . . 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 3023 . 2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( G `
 ( F  |`  suc  B ) ) )
72eleq1d 2508 . . . 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 3023 . . 3  |-  ( suc 
B  e.  X  -> 
( F  |`  suc  B
)  e.  _V )
10 noel 3617 . . . . . . 7  |-  -.  B  e.  (/)
11 dmeq 5099 . . . . . . . . 9  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  dom  (/) )
12 dm0 5112 . . . . . . . . 9  |-  dom  (/)  =  (/)
1311, 12syl6eq 2490 . . . . . . . 8  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  (/) )
14 tz7.44.5 . . . . . . . . . . . . 13  |-  Ord  X
15 ordsson 4799 . . . . . . . . . . . . 13  |-  ( Ord 
X  ->  X  C_  On )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  X  C_  On
17 ordtr 4624 . . . . . . . . . . . . . 14  |-  ( Ord 
X  ->  Tr  X
)
1814, 17ax-mp 5 . . . . . . . . . . . . 13  |-  Tr  X
19 trsuc 4695 . . . . . . . . . . . . 13  |-  ( ( Tr  X  /\  suc  B  e.  X )  ->  B  e.  X )
2018, 19mpan 653 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  B  e.  X )
2116, 20sseldi 3332 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  ->  B  e.  On )
22 sucidg 4688 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  e.  suc  B )
2321, 22syl 16 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  B  e.  suc  B )
24 dmres 5196 . . . . . . . . . . 11  |-  dom  ( F  |`  suc  B )  =  ( suc  B  i^i  dom  F )
25 ordelss 4626 . . . . . . . . . . . . . 14  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  C_  X )
2614, 25mpan 653 . . . . . . . . . . . . 13  |-  ( suc 
B  e.  X  ->  suc  B  C_  X )
27 tz7.44.4 . . . . . . . . . . . . . 14  |-  F  Fn  X
28 fndm 5573 . . . . . . . . . . . . . 14  |-  ( F  Fn  X  ->  dom  F  =  X )
2927, 28ax-mp 5 . . . . . . . . . . . . 13  |-  dom  F  =  X
3026, 29syl6sseqr 3381 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  suc  B  C_  dom  F )
31 df-ss 3320 . . . . . . . . . . . 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 2486 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  dom  ( F  |`  suc  B
)  =  suc  B
)
3423, 33eleqtrrd 2519 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  B  e.  dom  ( F  |`  suc  B ) )
35 eleq2 2503 . . . . . . . . 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 31 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B )  =  (/)  ->  B  e.  (/) ) )
3810, 37mtoi 172 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  ( F  |`  suc  B
)  =  (/) )
39 iffalse 3770 . . . . . 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 16 . . . . 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 4851 . . . . . . . 8  |-  ( B  e.  On  ->  -.  Lim  suc  B )
4221, 41syl 16 . . . . . . 7  |-  ( suc 
B  e.  X  ->  -.  Lim  suc  B )
43 limeq 4622 . . . . . . . 8  |-  ( dom  ( F  |`  suc  B
)  =  suc  B  ->  ( Lim  dom  ( F  |`  suc  B )  <->  Lim  suc  B ) )
4433, 43syl 16 . . . . . . 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 3770 . . . . . 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 16 . . . . 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 4050 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  U. suc  B )
49 eloni 4620 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
50 ordunisuc 4841 . . . . . . . . . . 11  |-  ( Ord 
B  ->  U. suc  B  =  B )
5149, 50syl 16 . . . . . . . . . 10  |-  ( B  e.  On  ->  U. suc  B  =  B )
5221, 51syl 16 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. suc  B  =  B )
5348, 52eqtrd 2474 . . . . . . . 8  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  B )
5453fveq2d 5761 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( ( F  |`  suc  B
) `  B )
)
55 fvres 5774 . . . . . . . 8  |-  ( B  e.  suc  B  -> 
( ( F  |`  suc  B ) `  B
)  =  ( F `
 B ) )
5623, 55syl 16 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  B
)  =  ( F `
 B ) )
5754, 56eqtrd 2474 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( F `  B ) )
5857fveq2d 5761 . . . . 5  |-  ( suc 
B  e.  X  -> 
( H `  (
( F  |`  suc  B
) `  U. dom  ( F  |`  suc  B ) ) )  =  ( H `  ( F `
 B ) ) )
5940, 47, 583eqtrd 2478 . . . 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 5771 . . . 4  |-  ( H `
 ( F `  B ) )  e. 
_V
6159, 60syl6eqel 2530 . . 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 2448 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  (
x  =  (/)  <->  ( F  |` 
suc  B )  =  (/) ) )
63 dmeq 5099 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  dom  x  =  dom  ( F  |`  suc  B ) )
64 limeq 4622 . . . . . . 7  |-  ( dom  x  =  dom  ( F  |`  suc  B )  ->  ( Lim  dom  x 
<->  Lim  dom  ( F  |` 
suc  B ) ) )
6563, 64syl 16 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( Lim  dom  x  <->  Lim  dom  ( F  |`  suc  B ) ) )
66 rneq 5124 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  ran  x  =  ran  ( F  |`  suc  B ) )
6766unieqd 4050 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  U. ran  x  =  U. ran  ( F  |`  suc  B ) )
68 fveq1 5756 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  x
) )
6963unieqd 4050 . . . . . . . . 9  |-  ( x  =  ( F  |`  suc  B )  ->  U. dom  x  =  U. dom  ( F  |`  suc  B ) )
7069fveq2d 5761 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
( F  |`  suc  B
) `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7168, 70eqtrd 2474 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7271fveq2d 5761 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( H `  ( x `  U. dom  x ) )  =  ( H `
 ( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B ) ) ) )
7365, 67, 72ifbieq12d 3785 . . . . 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 3783 . . . 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 5833 . . 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 644 . 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 2478 1  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( H `
 ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727   _Vcvv 2962    i^i cin 3305    C_ wss 3306   (/)c0 3613   ifcif 3763   U.cuni 4039    e. cmpt 4291   Tr wtr 4327   Ord word 4609   Oncon0 4610   Lim wlim 4611   suc csuc 4612   dom cdm 4907   ran crn 4908    |` cres 4909    Fn wfn 5478   ` cfv 5483
This theorem is referenced by:  rdgsucg  6710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491
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