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Theorem tz7.44-2 6966
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 5792 . . . 4  |-  ( y  =  suc  B  -> 
( F `  y
)  =  ( F `
 suc  B )
)
2 reseq2 5206 . . . . 5  |-  ( y  =  suc  B  -> 
( F  |`  y
)  =  ( F  |`  suc  B ) )
32fveq2d 5796 . . . 4  |-  ( y  =  suc  B  -> 
( G `  ( F  |`  y ) )  =  ( G `  ( F  |`  suc  B
) ) )
41, 3eqeq12d 2473 . . 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 3135 . 2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( G `
 ( F  |`  suc  B ) ) )
72eleq1d 2520 . . . 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 3135 . . 3  |-  ( suc 
B  e.  X  -> 
( F  |`  suc  B
)  e.  _V )
10 noel 3742 . . . . . . 7  |-  -.  B  e.  (/)
11 dmeq 5141 . . . . . . . . 9  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  dom  (/) )
12 dm0 5154 . . . . . . . . 9  |-  dom  (/)  =  (/)
1311, 12syl6eq 2508 . . . . . . . 8  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  ( F  |`  suc  B )  =  (/) )
14 tz7.44.5 . . . . . . . . . . . . 13  |-  Ord  X
15 ordsson 6504 . . . . . . . . . . . . 13  |-  ( Ord 
X  ->  X  C_  On )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  X  C_  On
17 ordtr 4834 . . . . . . . . . . . . . 14  |-  ( Ord 
X  ->  Tr  X
)
1814, 17ax-mp 5 . . . . . . . . . . . . 13  |-  Tr  X
19 trsuc 4904 . . . . . . . . . . . . 13  |-  ( ( Tr  X  /\  suc  B  e.  X )  ->  B  e.  X )
2018, 19mpan 670 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  B  e.  X )
2116, 20sseldi 3455 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  ->  B  e.  On )
22 sucidg 4898 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  e.  suc  B )
2321, 22syl 16 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  B  e.  suc  B )
24 dmres 5232 . . . . . . . . . . 11  |-  dom  ( F  |`  suc  B )  =  ( suc  B  i^i  dom  F )
25 ordelss 4836 . . . . . . . . . . . . . 14  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  C_  X )
2614, 25mpan 670 . . . . . . . . . . . . 13  |-  ( suc 
B  e.  X  ->  suc  B  C_  X )
27 tz7.44.4 . . . . . . . . . . . . . 14  |-  F  Fn  X
28 fndm 5611 . . . . . . . . . . . . . 14  |-  ( F  Fn  X  ->  dom  F  =  X )
2927, 28ax-mp 5 . . . . . . . . . . . . 13  |-  dom  F  =  X
3026, 29syl6sseqr 3504 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  suc  B  C_  dom  F )
31 df-ss 3443 . . . . . . . . . . . 12  |-  ( suc 
B  C_  dom  F  <->  ( suc  B  i^i  dom  F )  =  suc  B )
3230, 31sylib 196 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  -> 
( suc  B  i^i  dom 
F )  =  suc  B )
3324, 32syl5eq 2504 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  dom  ( F  |`  suc  B
)  =  suc  B
)
3423, 33eleqtrrd 2542 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  B  e.  dom  ( F  |`  suc  B ) )
35 eleq2 2524 . . . . . . . . 9  |-  ( dom  ( F  |`  suc  B
)  =  (/)  ->  ( B  e.  dom  ( F  |`  suc  B )  <->  B  e.  (/) ) )
3634, 35syl5ibcom 220 . . . . . . . 8  |-  ( suc 
B  e.  X  -> 
( dom  ( F  |` 
suc  B )  =  (/)  ->  B  e.  (/) ) )
3713, 36syl5 32 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B )  =  (/)  ->  B  e.  (/) ) )
3810, 37mtoi 178 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  ( F  |`  suc  B
)  =  (/) )
39 iffalse 3900 . . . . . 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 6556 . . . . . . . 8  |-  ( B  e.  On  ->  -.  Lim  suc  B )
4221, 41syl 16 . . . . . . 7  |-  ( suc 
B  e.  X  ->  -.  Lim  suc  B )
43 limeq 4832 . . . . . . . 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 301 . . . . . 6  |-  ( suc 
B  e.  X  ->  -.  Lim  dom  ( F  |` 
suc  B ) )
46 iffalse 3900 . . . . . 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 4202 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  U. suc  B )
49 eloni 4830 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
50 ordunisuc 6546 . . . . . . . . . . 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 2492 . . . . . . . 8  |-  ( suc 
B  e.  X  ->  U. dom  ( F  |`  suc  B )  =  B )
5453fveq2d 5796 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( ( F  |`  suc  B
) `  B )
)
55 fvres 5806 . . . . . . . 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 2492 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B
) )  =  ( F `  B ) )
5857fveq2d 5796 . . . . 5  |-  ( suc 
B  e.  X  -> 
( H `  (
( F  |`  suc  B
) `  U. dom  ( F  |`  suc  B ) ) )  =  ( H `  ( F `
 B ) ) )
5940, 47, 583eqtrd 2496 . . . 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 5802 . . . 4  |-  ( H `
 ( F `  B ) )  e. 
_V
6159, 60syl6eqel 2547 . . 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 2455 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  (
x  =  (/)  <->  ( F  |` 
suc  B )  =  (/) ) )
63 dmeq 5141 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  dom  x  =  dom  ( F  |`  suc  B ) )
64 limeq 4832 . . . . . . 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 5166 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  ran  x  =  ran  ( F  |`  suc  B ) )
6766unieqd 4202 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  U. ran  x  =  U. ran  ( F  |`  suc  B ) )
68 fveq1 5791 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  x
) )
6963unieqd 4202 . . . . . . . . 9  |-  ( x  =  ( F  |`  suc  B )  ->  U. dom  x  =  U. dom  ( F  |`  suc  B ) )
7069fveq2d 5796 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
( F  |`  suc  B
) `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7168, 70eqtrd 2492 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  ( F  |`  suc  B ) ) )
7271fveq2d 5796 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( H `  ( x `  U. dom  x ) )  =  ( H `
 ( ( F  |`  suc  B ) `  U. dom  ( F  |`  suc  B ) ) ) )
7365, 67, 72ifbieq12d 3917 . . . . 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 3915 . . . 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 5874 . . 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 661 . 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 2496 1  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( H `
 ( F `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3071    i^i cin 3428    C_ wss 3429   (/)c0 3738   ifcif 3892   U.cuni 4192    |-> cmpt 4451   Tr wtr 4486   Ord word 4819   Oncon0 4820   Lim wlim 4821   suc csuc 4822   dom cdm 4941   ran crn 4942    |` cres 4943    Fn wfn 5514   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527
This theorem is referenced by:  rdgsucg  6982
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