MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itunitc Unicode version

Theorem itunitc 8047
Description: The union of all union iterates creates the transitive closure; compare trcl 7410. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Hypothesis
Ref Expression
ituni.u  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
Assertion
Ref Expression
itunitc  |-  ( TC
`  A )  = 
U. ran  ( U `  A )
Distinct variable group:    x, A, y
Allowed substitution hints:    U( x, y)

Proof of Theorem itunitc
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
2 fveq2 5525 . . . . . 6  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
32rneqd 4906 . . . . 5  |-  ( a  =  A  ->  ran  ( U `  a )  =  ran  ( U `
 A ) )
43unieqd 3838 . . . 4  |-  ( a  =  A  ->  U. ran  ( U `  a )  =  U. ran  ( U `  A )
)
51, 4eqeq12d 2297 . . 3  |-  ( a  =  A  ->  (
( TC `  a
)  =  U. ran  ( U `  a )  <-> 
( TC `  A
)  =  U. ran  ( U `  A ) ) )
6 vex 2791 . . . . . . 7  |-  a  e. 
_V
7 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87ituni0 8044 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
96, 8ax-mp 8 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  =  a
10 fvssunirn 5551 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  C_  U. ran  ( U `  a )
119, 10eqsstr3i 3209 . . . . 5  |-  a  C_  U.
ran  ( U `  a )
12 dftr3 4117 . . . . . 6  |-  ( Tr 
U. ran  ( U `  a )  <->  A. b  e.  U. ran  ( U `
 a ) b 
C_  U. ran  ( U `
 a ) )
137itunifn 8043 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
14 fnunirn 5778 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  (
b  e.  U. ran  ( U `  a )  <->  E. c  e.  om  b  e.  ( ( U `  a ) `  c ) ) )
156, 13, 14mp2b 9 . . . . . . 7  |-  ( b  e.  U. ran  ( U `  a )  <->  E. c  e.  om  b  e.  ( ( U `  a ) `  c
) )
16 elssuni 3855 . . . . . . . . 9  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ( ( U `
 a ) `  c ) )
177itunisuc 8045 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
18 fvssunirn 5551 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  C_ 
U. ran  ( U `  a )
1917, 18eqsstr3i 3209 . . . . . . . . 9  |-  U. (
( U `  a
) `  c )  C_ 
U. ran  ( U `  a )
2016, 19syl6ss 3191 . . . . . . . 8  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ran  ( U `  a ) )
2120rexlimivw 2663 . . . . . . 7  |-  ( E. c  e.  om  b  e.  ( ( U `  a ) `  c
)  ->  b  C_  U.
ran  ( U `  a ) )
2215, 21sylbi 187 . . . . . 6  |-  ( b  e.  U. ran  ( U `  a )  ->  b  C_  U. ran  ( U `  a )
)
2312, 22mprgbir 2613 . . . . 5  |-  Tr  U. ran  ( U `  a
)
24 tcmin 7426 . . . . . 6  |-  ( a  e.  _V  ->  (
( a  C_  U. ran  ( U `  a )  /\  Tr  U. ran  ( U `  a ) )  ->  ( TC `  a )  C_  U. ran  ( U `  a ) ) )
256, 24ax-mp 8 . . . . 5  |-  ( ( a  C_  U. ran  ( U `  a )  /\  Tr  U. ran  ( U `  a )
)  ->  ( TC `  a )  C_  U. ran  ( U `  a ) )
2611, 23, 25mp2an 653 . . . 4  |-  ( TC
`  a )  C_  U.
ran  ( U `  a )
27 unissb 3857 . . . . 5  |-  ( U. ran  ( U `  a
)  C_  ( TC `  a )  <->  A. b  e.  ran  ( U `  a ) b  C_  ( TC `  a ) )
28 fvelrnb 5570 . . . . . . 7  |-  ( ( U `  a )  Fn  om  ->  (
b  e.  ran  ( U `  a )  <->  E. c  e.  om  (
( U `  a
) `  c )  =  b ) )
296, 13, 28mp2b 9 . . . . . 6  |-  ( b  e.  ran  ( U `
 a )  <->  E. c  e.  om  ( ( U `
 a ) `  c )  =  b )
307itunitc1 8046 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  C_  ( TC `  a )
3130a1i 10 . . . . . . . 8  |-  ( c  e.  om  ->  (
( U `  a
) `  c )  C_  ( TC `  a
) )
32 sseq1 3199 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  =  b  ->  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  <->  b  C_  ( TC `  a ) ) )
3331, 32syl5ibcom 211 . . . . . . 7  |-  ( c  e.  om  ->  (
( ( U `  a ) `  c
)  =  b  -> 
b  C_  ( TC `  a ) ) )
3433rexlimiv 2661 . . . . . 6  |-  ( E. c  e.  om  (
( U `  a
) `  c )  =  b  ->  b  C_  ( TC `  a ) )
3529, 34sylbi 187 . . . . 5  |-  ( b  e.  ran  ( U `
 a )  -> 
b  C_  ( TC `  a ) )
3627, 35mprgbir 2613 . . . 4  |-  U. ran  ( U `  a ) 
C_  ( TC `  a )
3726, 36eqssi 3195 . . 3  |-  ( TC
`  a )  = 
U. ran  ( U `  a )
385, 37vtoclg 2843 . 2  |-  ( A  e.  _V  ->  ( TC `  A )  = 
U. ran  ( U `  A ) )
39 rn0 4936 . . . . 5  |-  ran  (/)  =  (/)
4039unieqi 3837 . . . 4  |-  U. ran  (/)  =  U. (/)
41 uni0 3854 . . . 4  |-  U. (/)  =  (/)
4240, 41eqtr2i 2304 . . 3  |-  (/)  =  U. ran  (/)
43 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
44 fvprc 5519 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544rneqd 4906 . . . 4  |-  ( -.  A  e.  _V  ->  ran  ( U `  A
)  =  ran  (/) )
4645unieqd 3838 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  ( U `  A )  =  U. ran  (/) )
4742, 43, 463eqtr4a 2341 . 2  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  U. ran  ( U `  A )
)
4838, 47pm2.61i 156 1  |-  ( TC
`  A )  = 
U. ran  ( U `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U.cuni 3827    e. cmpt 4077   Tr wtr 4113   suc csuc 4394   omcom 4656   ran crn 4690    |` cres 4691    Fn wfn 5250   ` cfv 5255   reccrdg 6422   TCctc 7421
This theorem is referenced by:  hsmexlem5  8056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-tc 7422
  Copyright terms: Public domain W3C validator