MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.44-2 Unicode version

Theorem tz7.44-2 6415
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 5485 . . . 4  |-  ( y  =  suc  B  -> 
( F `  y
)  =  ( F `
 suc  B )
)
2 reseq2 4949 . . . . 5  |-  ( y  =  suc  B  -> 
( F  |`  y
)  =  ( F  |`  suc  B ) )
32fveq2d 5489 . . . 4  |-  ( y  =  suc  B  -> 
( G `  ( F  |`  y ) )  =  ( G `  ( F  |`  suc  B
) ) )
41, 3eqeq12d 2298 . . 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 2850 . 2  |-  ( suc 
B  e.  X  -> 
( F `  suc  B )  =  ( G `
 ( F  |`  suc  B ) ) )
72eleq1d 2350 . . . 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 2850 . . 3  |-  ( suc 
B  e.  X  -> 
( F  |`  suc  B
)  e.  _V )
10 noel 3460 . . . . . . 7  |-  -.  B  e.  (/)
11 dmeq 4878 . . . . . . . . 9  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  (  F  |`  suc  B )  =  dom  (/) )
12 dm0 4891 . . . . . . . . 9  |-  dom  (/)  =  (/)
1311, 12syl6eq 2332 . . . . . . . 8  |-  ( ( F  |`  suc  B )  =  (/)  ->  dom  (  F  |`  suc  B )  =  (/) )
14 tz7.44.5 . . . . . . . . . . . . 13  |-  Ord  X
15 ordsson 4580 . . . . . . . . . . . . 13  |-  ( Ord 
X  ->  X  C_  On )
1614, 15ax-mp 10 . . . . . . . . . . . 12  |-  X  C_  On
17 ordtr 4405 . . . . . . . . . . . . . 14  |-  ( Ord 
X  ->  Tr  X
)
1814, 17ax-mp 10 . . . . . . . . . . . . 13  |-  Tr  X
19 trsuc 4475 . . . . . . . . . . . . 13  |-  ( ( Tr  X  /\  suc  B  e.  X )  ->  B  e.  X )
2018, 19mpan 654 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  B  e.  X )
2116, 20sseldi 3179 . . . . . . . . . . 11  |-  ( suc 
B  e.  X  ->  B  e.  On )
22 sucidg 4469 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  e.  suc  B )
2321, 22syl 17 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  B  e.  suc  B )
24 dmres 4975 . . . . . . . . . . 11  |-  dom  (  F  |`  suc  B )  =  ( suc  B  i^i  dom  F )
25 ordelss 4407 . . . . . . . . . . . . . 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 5308 . . . . . . . . . . . . . 14  |-  ( F  Fn  X  ->  dom  F  =  X )
2927, 28ax-mp 10 . . . . . . . . . . . . 13  |-  dom  F  =  X
3026, 29syl6sseqr 3226 . . . . . . . . . . . 12  |-  ( suc 
B  e.  X  ->  suc  B  C_  dom  F )
31 df-ss 3167 . . . . . . . . . . . 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 2328 . . . . . . . . . 10  |-  ( suc 
B  e.  X  ->  dom  (  F  |`  suc  B
)  =  suc  B
)
3423, 33eleqtrrd 2361 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  B  e.  dom  (  F  |`  suc  B ) )
35 eleq2 2345 . . . . . . . . 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 3573 . . . . . 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 4632 . . . . . . . 8  |-  ( B  e.  On  ->  -.  Lim  suc  B )
4221, 41syl 17 . . . . . . 7  |-  ( suc 
B  e.  X  ->  -.  Lim  suc  B )
43 limeq 4403 . . . . . . . 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 3573 . . . . . 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 3839 . . . . . . . . 9  |-  ( suc 
B  e.  X  ->  U. dom  (  F  |`  suc  B )  =  U. suc  B )
49 eloni 4401 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
50 ordunisuc 4622 . . . . . . . . . . 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 2316 . . . . . . . 8  |-  ( suc 
B  e.  X  ->  U. dom  (  F  |`  suc  B )  =  B )
5453fveq2d 5489 . . . . . . 7  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) )  =  ( ( F  |`  suc  B
) `  B )
)
55 fvres 5502 . . . . . . . 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 2316 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B
) )  =  ( F `  B ) )
5857fveq2d 5489 . . . . 5  |-  ( suc 
B  e.  X  -> 
( H `  (
( F  |`  suc  B
) `  U. dom  (  F  |`  suc  B ) ) )  =  ( H `  ( F `
 B ) ) )
5940, 47, 583eqtrd 2320 . . . 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 5499 . . . 4  |-  ( H `
 ( F `  B ) )  e. 
_V
6159, 60syl6eqel 2372 . . 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 2290 . . . . 5  |-  ( x  =  ( F  |`  suc  B )  ->  (
x  =  (/)  <->  ( F  |` 
suc  B )  =  (/) ) )
63 dmeq 4878 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  dom  x  =  dom  (  F  |`  suc  B ) )
64 limeq 4403 . . . . . . 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 4903 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  ran  x  =  ran  (  F  |`  suc  B ) )
6766unieqd 3839 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  U. ran  x  =  U. ran  (  F  |`  suc  B ) )
68 fveq1 5484 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  x
) )
6963unieqd 3839 . . . . . . . . 9  |-  ( x  =  ( F  |`  suc  B )  ->  U. dom  x  =  U. dom  (  F  |`  suc  B ) )
7069fveq2d 5489 . . . . . . . 8  |-  ( x  =  ( F  |`  suc  B )  ->  (
( F  |`  suc  B
) `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  (  F  |`  suc  B ) ) )
7168, 70eqtrd 2316 . . . . . . 7  |-  ( x  =  ( F  |`  suc  B )  ->  (
x `  U. dom  x
)  =  ( ( F  |`  suc  B ) `
 U. dom  (  F  |`  suc  B ) ) )
7271fveq2d 5489 . . . . . 6  |-  ( x  =  ( F  |`  suc  B )  ->  ( H `  ( x `  U. dom  x ) )  =  ( H `
 ( ( F  |`  suc  B ) `  U. dom  (  F  |`  suc  B ) ) ) )
7365, 67, 72ifbieq12d 3588 . . . . 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 3586 . . . 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 5561 . . 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 2320 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 1628    e. wcel 1688   _Vcvv 2789    i^i cin 3152    C_ wss 3153   (/)c0 3456   ifcif 3566   U.cuni 3828    e. cmpt 4078   Tr wtr 4114   Ord word 4390   Oncon0 4391   Lim wlim 4392   suc csuc 4393   dom cdm 4688   ran crn 4689    |` cres 4690    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  rdgsucg  6431
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
  Copyright terms: Public domain W3C validator