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Theorem itunitc 8857
Description: The union of all union iterates creates the transitive closure; compare trcl 8219. (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 5880 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
2 fveq2 5880 . . . . . 6  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
32rneqd 5080 . . . . 5  |-  ( a  =  A  ->  ran  ( U `  a )  =  ran  ( U `
 A ) )
43unieqd 4228 . . . 4  |-  ( a  =  A  ->  U. ran  ( U `  a )  =  U. ran  ( U `  A )
)
51, 4eqeq12d 2445 . . 3  |-  ( a  =  A  ->  (
( TC `  a
)  =  U. ran  ( U `  a )  <-> 
( TC `  A
)  =  U. ran  ( U `  A ) ) )
6 vex 3085 . . . . . . 7  |-  a  e. 
_V
7 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87ituni0 8854 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
96, 8ax-mp 5 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  =  a
10 fvssunirn 5903 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  C_  U. ran  ( U `  a )
119, 10eqsstr3i 3497 . . . . 5  |-  a  C_  U.
ran  ( U `  a )
12 dftr3 4521 . . . . . 6  |-  ( Tr 
U. ran  ( U `  a )  <->  A. b  e.  U. ran  ( U `
 a ) b 
C_  U. ran  ( U `
 a ) )
137itunifn 8853 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
14 fnunirn 6172 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  (
b  e.  U. ran  ( U `  a )  <->  E. c  e.  om  b  e.  ( ( U `  a ) `  c ) ) )
156, 13, 14mp2b 10 . . . . . . 7  |-  ( b  e.  U. ran  ( U `  a )  <->  E. c  e.  om  b  e.  ( ( U `  a ) `  c
) )
16 elssuni 4247 . . . . . . . . 9  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ( ( U `
 a ) `  c ) )
177itunisuc 8855 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
18 fvssunirn 5903 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  C_ 
U. ran  ( U `  a )
1917, 18eqsstr3i 3497 . . . . . . . . 9  |-  U. (
( U `  a
) `  c )  C_ 
U. ran  ( U `  a )
2016, 19syl6ss 3478 . . . . . . . 8  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ran  ( U `  a ) )
2120rexlimivw 2915 . . . . . . 7  |-  ( E. c  e.  om  b  e.  ( ( U `  a ) `  c
)  ->  b  C_  U.
ran  ( U `  a ) )
2215, 21sylbi 199 . . . . . 6  |-  ( b  e.  U. ran  ( U `  a )  ->  b  C_  U. ran  ( U `  a )
)
2312, 22mprgbir 2790 . . . . 5  |-  Tr  U. ran  ( U `  a
)
24 tcmin 8232 . . . . . 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 5 . . . . 5  |-  ( ( a  C_  U. ran  ( U `  a )  /\  Tr  U. ran  ( U `  a )
)  ->  ( TC `  a )  C_  U. ran  ( U `  a ) )
2611, 23, 25mp2an 677 . . . 4  |-  ( TC
`  a )  C_  U.
ran  ( U `  a )
27 unissb 4249 . . . . 5  |-  ( U. ran  ( U `  a
)  C_  ( TC `  a )  <->  A. b  e.  ran  ( U `  a ) b  C_  ( TC `  a ) )
28 fvelrnb 5927 . . . . . . 7  |-  ( ( U `  a )  Fn  om  ->  (
b  e.  ran  ( U `  a )  <->  E. c  e.  om  (
( U `  a
) `  c )  =  b ) )
296, 13, 28mp2b 10 . . . . . 6  |-  ( b  e.  ran  ( U `
 a )  <->  E. c  e.  om  ( ( U `
 a ) `  c )  =  b )
307itunitc1 8856 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  C_  ( TC `  a )
3130a1i 11 . . . . . . . 8  |-  ( c  e.  om  ->  (
( U `  a
) `  c )  C_  ( TC `  a
) )
32 sseq1 3487 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  =  b  ->  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  <->  b  C_  ( TC `  a ) ) )
3331, 32syl5ibcom 224 . . . . . . 7  |-  ( c  e.  om  ->  (
( ( U `  a ) `  c
)  =  b  -> 
b  C_  ( TC `  a ) ) )
3433rexlimiv 2912 . . . . . 6  |-  ( E. c  e.  om  (
( U `  a
) `  c )  =  b  ->  b  C_  ( TC `  a ) )
3529, 34sylbi 199 . . . . 5  |-  ( b  e.  ran  ( U `
 a )  -> 
b  C_  ( TC `  a ) )
3627, 35mprgbir 2790 . . . 4  |-  U. ran  ( U `  a ) 
C_  ( TC `  a )
3726, 36eqssi 3482 . . 3  |-  ( TC
`  a )  = 
U. ran  ( U `  a )
385, 37vtoclg 3140 . 2  |-  ( A  e.  _V  ->  ( TC `  A )  = 
U. ran  ( U `  A ) )
39 rn0 5104 . . . . 5  |-  ran  (/)  =  (/)
4039unieqi 4227 . . . 4  |-  U. ran  (/)  =  U. (/)
41 uni0 4245 . . . 4  |-  U. (/)  =  (/)
4240, 41eqtr2i 2453 . . 3  |-  (/)  =  U. ran  (/)
43 fvprc 5874 . . 3  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
44 fvprc 5874 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544rneqd 5080 . . . 4  |-  ( -.  A  e.  _V  ->  ran  ( U `  A
)  =  ran  (/) )
4645unieqd 4228 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  ( U `  A )  =  U. ran  (/) )
4742, 43, 463eqtr4a 2490 . 2  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  U. ran  ( U `  A )
)
4838, 47pm2.61i 168 1  |-  ( TC
`  A )  = 
U. ran  ( U `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   E.wrex 2777   _Vcvv 3082    C_ wss 3438   (/)c0 3763   U.cuni 4218    |-> cmpt 4481   Tr wtr 4517   ran crn 4853    |` cres 4854   suc csuc 5443    Fn wfn 5595   ` cfv 5600   omcom 6705   reccrdg 7137   TCctc 8227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596  ax-inf2 8154
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-pss 3454  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4219  df-int 4255  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-tr 4518  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6706  df-wrecs 7038  df-recs 7100  df-rdg 7138  df-tc 8228
This theorem is referenced by:  hsmexlem5  8866
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