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Theorem tz7.44-2 6651
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 5714 . . . 4  |-  ( y  =  suc  B  -> 
( F `  y
)  =  ( F `
 suc  B )
)
2 reseq2 5127 . . . . 5  |-  ( y  =  suc  B  -> 
( F  |`  y
)  =  ( F  |`  suc  B ) )
32fveq2d 5718 . . . 4  |-  ( y  =  suc  B  -> 
( G `  ( F  |`  y ) )  =  ( G `  ( F  |`  suc  B
) ) )
41, 3eqeq12d 2444 . . 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 3004 . 2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( G `
 ( F  |`  suc  B ) ) )
72eleq1d 2496 . . . 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 3004 . . 3  |-  ( suc 
B  e.  X  -> 
( F  |`  suc  B
)  e.  _V )
10 noel 3619 . . . . . . 7  |-  -.  B  e.  (/)
11 dmeq 5056 . . . . . . . . 9  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  dom  (/) )
12 dm0 5069 . . . . . . . . 9  |-  dom  (/)  =  (/)
1311, 12syl6eq 2478 . . . . . . . 8  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  (/) )
14 tz7.44.5 . . . . . . . . . . . . 13  |-  Ord  X
15 ordsson 4756 . . . . . . . . . . . . 13  |-  ( Ord 
X  ->  X  C_  On )
1614, 15ax-mp 8 . . . . . . . . . . . 12  |-  X  C_  On
17 ordtr 4582 . . . . . . . . . . . . . 14  |-  ( Ord 
X  ->  Tr  X
)
1814, 17ax-mp 8 . . . . . . . . . . . . 13  |-  Tr  X
19 trsuc 4652 . . . . . . . . . . . . 13  |-  ( ( Tr  X  /\  suc  B  e.  X )  ->  B  e.  X )
2018, 19mpan 652 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  B  e.  X )
2116, 20sseldi 3333 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  ->  B  e.  On )
22 sucidg 4646 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  e.  suc  B )
2321, 22syl 16 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  B  e.  suc  B )
24 dmres 5153 . . . . . . . . . . 11  |-  dom  ( F  |`  suc  B )  =  ( suc  B  i^i  dom  F )
25 ordelss 4584 . . . . . . . . . . . . . 14  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  C_  X )
2614, 25mpan 652 . . . . . . . . . . . . 13  |-  ( suc 
B  e.  X  ->  suc  B  C_  X )
27 tz7.44.4 . . . . . . . . . . . . . 14  |-  F  Fn  X
28 fndm 5530 . . . . . . . . . . . . . 14  |-  ( F  Fn  X  ->  dom  F  =  X )
2927, 28ax-mp 8 . . . . . . . . . . . . 13  |-  dom  F  =  X
3026, 29syl6sseqr 3382 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  suc  B  C_  dom  F )
31 df-ss 3321 . . . . . . . . . . . 12  |-  ( suc 
B  C_  dom  F  <->  ( suc  B  i^i  dom  F )  =  suc  B )
3230, 31sylib 189 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  -> 
( suc  B  i^i  dom 
F )  =  suc  B )
3324, 32syl5eq 2474 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  dom  ( F  |`  suc  B
)  =  suc  B
)
3423, 33eleqtrrd 2507 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  B  e.  dom  ( F  |`  suc  B ) )
35 eleq2 2491 . . . . . . . . 9  |-  ( dom  ( F  |`  suc  B
)  =  (/)  ->  ( B  e.  dom  ( F  |`  suc  B )  <->  B  e.  (/) ) )
3634, 35syl5ibcom 212 . . . . . . . 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 3733 . . . . . 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 4808 . . . . . . . 8  |-  ( B  e.  On  ->  -.  Lim  suc  B )
4221, 41syl 16 . . . . . . 7  |-  ( suc 
B  e.  X  ->  -.  Lim  suc  B )
43 limeq 4580 . . . . . . . 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 293 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  Lim  dom  ( F  |` 
suc  B ) )
46 iffalse 3733 . . . . . 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 4013 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  U. suc  B )
49 eloni 4578 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
50 ordunisuc 4798 . . . . . . . . . . 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 2462 . . . . . . . 8  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  B )
5453fveq2d 5718 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( ( F  |`  suc  B
) `  B )
)
55 fvres 5731 . . . . . . . 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 2462 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( F `  B ) )
5857fveq2d 5718 . . . . 5  |-  ( suc 
B  e.  X  -> 
( H `  (
( F  |`  suc  B
) `  U. dom  ( F  |`  suc  B ) ) )  =  ( H `  ( F `
 B ) ) )
5940, 47, 583eqtrd 2466 . . . 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 5728 . . . 4  |-  ( H `
 ( F `  B ) )  e. 
_V
6159, 60syl6eqel 2518 . . 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 2436 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  (
x  =  (/)  <->  ( F  |` 
suc  B )  =  (/) ) )
63 dmeq 5056 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  dom  x  =  dom  ( F  |`  suc  B ) )
64 limeq 4580 . . . . . . 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 5081 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  ran  x  =  ran  ( F  |`  suc  B ) )
6766unieqd 4013 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  U. ran  x  =  U. ran  ( F  |`  suc  B ) )
68 fveq1 5713 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  x
) )
6963unieqd 4013 . . . . . . . . 9  |-  ( x  =  ( F  |`  suc  B )  ->  U. dom  x  =  U. dom  ( F  |`  suc  B ) )
7069fveq2d 5718 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
( F  |`  suc  B
) `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7168, 70eqtrd 2462 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7271fveq2d 5718 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( H `  ( x `  U. dom  x ) )  =  ( H `
 ( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B ) ) ) )
7365, 67, 72ifbieq12d 3748 . . . . 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 3746 . . . 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 5790 . . 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 643 . 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 2466 1  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( H `
 ( F `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2943    i^i cin 3306    C_ wss 3307   (/)c0 3615   ifcif 3726   U.cuni 4002    e. cmpt 4253   Tr wtr 4289   Ord word 4567   Oncon0 4568   Lim wlim 4569   suc csuc 4570   dom cdm 4864   ran crn 4865    |` cres 4866    Fn wfn 5435   ` cfv 5440
This theorem is referenced by:  rdgsucg  6667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-iota 5404  df-fun 5442  df-fn 5443  df-fv 5448
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