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Theorem itunitc 8792
Description: The union of all union iterates creates the transitive closure; compare trcl 8150. (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 5859 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
2 fveq2 5859 . . . . . 6  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
32rneqd 5223 . . . . 5  |-  ( a  =  A  ->  ran  ( U `  a )  =  ran  ( U `
 A ) )
43unieqd 4250 . . . 4  |-  ( a  =  A  ->  U. ran  ( U `  a )  =  U. ran  ( U `  A )
)
51, 4eqeq12d 2484 . . 3  |-  ( a  =  A  ->  (
( TC `  a
)  =  U. ran  ( U `  a )  <-> 
( TC `  A
)  =  U. ran  ( U `  A ) ) )
6 vex 3111 . . . . . . 7  |-  a  e. 
_V
7 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87ituni0 8789 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
96, 8ax-mp 5 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  =  a
10 fvssunirn 5882 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  C_  U. ran  ( U `  a )
119, 10eqsstr3i 3530 . . . . 5  |-  a  C_  U.
ran  ( U `  a )
12 dftr3 4539 . . . . . 6  |-  ( Tr 
U. ran  ( U `  a )  <->  A. b  e.  U. ran  ( U `
 a ) b 
C_  U. ran  ( U `
 a ) )
137itunifn 8788 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
14 fnunirn 6146 . . . . . . . 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 4270 . . . . . . . . 9  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ( ( U `
 a ) `  c ) )
177itunisuc 8790 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
18 fvssunirn 5882 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  C_ 
U. ran  ( U `  a )
1917, 18eqsstr3i 3530 . . . . . . . . 9  |-  U. (
( U `  a
) `  c )  C_ 
U. ran  ( U `  a )
2016, 19syl6ss 3511 . . . . . . . 8  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ran  ( U `  a ) )
2120rexlimivw 2947 . . . . . . 7  |-  ( E. c  e.  om  b  e.  ( ( U `  a ) `  c
)  ->  b  C_  U.
ran  ( U `  a ) )
2215, 21sylbi 195 . . . . . 6  |-  ( b  e.  U. ran  ( U `  a )  ->  b  C_  U. ran  ( U `  a )
)
2312, 22mprgbir 2823 . . . . 5  |-  Tr  U. ran  ( U `  a
)
24 tcmin 8163 . . . . . 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 672 . . . 4  |-  ( TC
`  a )  C_  U.
ran  ( U `  a )
27 unissb 4272 . . . . 5  |-  ( U. ran  ( U `  a
)  C_  ( TC `  a )  <->  A. b  e.  ran  ( U `  a ) b  C_  ( TC `  a ) )
28 fvelrnb 5908 . . . . . . 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 8791 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  C_  ( TC `  a )
3130a1i 11 . . . . . . . 8  |-  ( c  e.  om  ->  (
( U `  a
) `  c )  C_  ( TC `  a
) )
32 sseq1 3520 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  =  b  ->  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  <->  b  C_  ( TC `  a ) ) )
3331, 32syl5ibcom 220 . . . . . . 7  |-  ( c  e.  om  ->  (
( ( U `  a ) `  c
)  =  b  -> 
b  C_  ( TC `  a ) ) )
3433rexlimiv 2944 . . . . . 6  |-  ( E. c  e.  om  (
( U `  a
) `  c )  =  b  ->  b  C_  ( TC `  a ) )
3529, 34sylbi 195 . . . . 5  |-  ( b  e.  ran  ( U `
 a )  -> 
b  C_  ( TC `  a ) )
3627, 35mprgbir 2823 . . . 4  |-  U. ran  ( U `  a ) 
C_  ( TC `  a )
3726, 36eqssi 3515 . . 3  |-  ( TC
`  a )  = 
U. ran  ( U `  a )
385, 37vtoclg 3166 . 2  |-  ( A  e.  _V  ->  ( TC `  A )  = 
U. ran  ( U `  A ) )
39 rn0 5247 . . . . 5  |-  ran  (/)  =  (/)
4039unieqi 4249 . . . 4  |-  U. ran  (/)  =  U. (/)
41 uni0 4267 . . . 4  |-  U. (/)  =  (/)
4240, 41eqtr2i 2492 . . 3  |-  (/)  =  U. ran  (/)
43 fvprc 5853 . . 3  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
44 fvprc 5853 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544rneqd 5223 . . . 4  |-  ( -.  A  e.  _V  ->  ran  ( U `  A
)  =  ran  (/) )
4645unieqd 4250 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  ( U `  A )  =  U. ran  (/) )
4742, 43, 463eqtr4a 2529 . 2  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  U. ran  ( U `  A )
)
4838, 47pm2.61i 164 1  |-  ( TC
`  A )  = 
U. ran  ( U `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108    C_ wss 3471   (/)c0 3780   U.cuni 4240    |-> cmpt 4500   Tr wtr 4535   suc csuc 4875   ran crn 4995    |` cres 4996    Fn wfn 5576   ` cfv 5581   omcom 6673   reccrdg 7067   TCctc 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-tc 8159
This theorem is referenced by:  hsmexlem5  8801
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