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Theorem itunitc 8063
Description: The union of all union iterates creates the transitive closure; compare trcl 7426. (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 5541 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
2 fveq2 5541 . . . . . 6  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
32rneqd 4922 . . . . 5  |-  ( a  =  A  ->  ran  ( U `  a )  =  ran  ( U `
 A ) )
43unieqd 3854 . . . 4  |-  ( a  =  A  ->  U. ran  ( U `  a )  =  U. ran  ( U `  A )
)
51, 4eqeq12d 2310 . . 3  |-  ( a  =  A  ->  (
( TC `  a
)  =  U. ran  ( U `  a )  <-> 
( TC `  A
)  =  U. ran  ( U `  A ) ) )
6 vex 2804 . . . . . . 7  |-  a  e. 
_V
7 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87ituni0 8060 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
96, 8ax-mp 8 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  =  a
10 fvssunirn 5567 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  C_  U. ran  ( U `  a )
119, 10eqsstr3i 3222 . . . . 5  |-  a  C_  U.
ran  ( U `  a )
12 dftr3 4133 . . . . . 6  |-  ( Tr 
U. ran  ( U `  a )  <->  A. b  e.  U. ran  ( U `
 a ) b 
C_  U. ran  ( U `
 a ) )
137itunifn 8059 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
14 fnunirn 5794 . . . . . . . 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 3871 . . . . . . . . 9  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ( ( U `
 a ) `  c ) )
177itunisuc 8061 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
18 fvssunirn 5567 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  C_ 
U. ran  ( U `  a )
1917, 18eqsstr3i 3222 . . . . . . . . 9  |-  U. (
( U `  a
) `  c )  C_ 
U. ran  ( U `  a )
2016, 19syl6ss 3204 . . . . . . . 8  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ran  ( U `  a ) )
2120rexlimivw 2676 . . . . . . 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 2626 . . . . 5  |-  Tr  U. ran  ( U `  a
)
24 tcmin 7442 . . . . . 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 3873 . . . . 5  |-  ( U. ran  ( U `  a
)  C_  ( TC `  a )  <->  A. b  e.  ran  ( U `  a ) b  C_  ( TC `  a ) )
28 fvelrnb 5586 . . . . . . 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 8062 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  C_  ( TC `  a )
3130a1i 10 . . . . . . . 8  |-  ( c  e.  om  ->  (
( U `  a
) `  c )  C_  ( TC `  a
) )
32 sseq1 3212 . . . . . . . 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 2674 . . . . . 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 2626 . . . 4  |-  U. ran  ( U `  a ) 
C_  ( TC `  a )
3726, 36eqssi 3208 . . 3  |-  ( TC
`  a )  = 
U. ran  ( U `  a )
385, 37vtoclg 2856 . 2  |-  ( A  e.  _V  ->  ( TC `  A )  = 
U. ran  ( U `  A ) )
39 rn0 4952 . . . . 5  |-  ran  (/)  =  (/)
4039unieqi 3853 . . . 4  |-  U. ran  (/)  =  U. (/)
41 uni0 3870 . . . 4  |-  U. (/)  =  (/)
4240, 41eqtr2i 2317 . . 3  |-  (/)  =  U. ran  (/)
43 fvprc 5535 . . 3  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
44 fvprc 5535 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544rneqd 4922 . . . 4  |-  ( -.  A  e.  _V  ->  ran  ( U `  A
)  =  ran  (/) )
4645unieqd 3854 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  ( U `  A )  =  U. ran  (/) )
4742, 43, 463eqtr4a 2354 . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U.cuni 3843    e. cmpt 4093   Tr wtr 4129   suc csuc 4410   omcom 4672   ran crn 4706    |` cres 4707    Fn wfn 5266   ` cfv 5271   reccrdg 6438   TCctc 7437
This theorem is referenced by:  hsmexlem5  8072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-tc 7438
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