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Theorem itunitc 8714
Description: The union of all union iterates creates the transitive closure; compare trcl 8072. (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 5774 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
2 fveq2 5774 . . . . . 6  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
32rneqd 5143 . . . . 5  |-  ( a  =  A  ->  ran  ( U `  a )  =  ran  ( U `
 A ) )
43unieqd 4173 . . . 4  |-  ( a  =  A  ->  U. ran  ( U `  a )  =  U. ran  ( U `  A )
)
51, 4eqeq12d 2404 . . 3  |-  ( a  =  A  ->  (
( TC `  a
)  =  U. ran  ( U `  a )  <-> 
( TC `  A
)  =  U. ran  ( U `  A ) ) )
6 vex 3037 . . . . . . 7  |-  a  e. 
_V
7 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
87ituni0 8711 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
96, 8ax-mp 5 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  =  a
10 fvssunirn 5797 . . . . . 6  |-  ( ( U `  a ) `
 (/) )  C_  U. ran  ( U `  a )
119, 10eqsstr3i 3448 . . . . 5  |-  a  C_  U.
ran  ( U `  a )
12 dftr3 4464 . . . . . 6  |-  ( Tr 
U. ran  ( U `  a )  <->  A. b  e.  U. ran  ( U `
 a ) b 
C_  U. ran  ( U `
 a ) )
137itunifn 8710 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
14 fnunirn 6066 . . . . . . . 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 4192 . . . . . . . . 9  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ( ( U `
 a ) `  c ) )
177itunisuc 8712 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
18 fvssunirn 5797 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  C_ 
U. ran  ( U `  a )
1917, 18eqsstr3i 3448 . . . . . . . . 9  |-  U. (
( U `  a
) `  c )  C_ 
U. ran  ( U `  a )
2016, 19syl6ss 3429 . . . . . . . 8  |-  ( b  e.  ( ( U `
 a ) `  c )  ->  b  C_ 
U. ran  ( U `  a ) )
2120rexlimivw 2871 . . . . . . 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 2746 . . . . 5  |-  Tr  U. ran  ( U `  a
)
24 tcmin 8085 . . . . . 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 670 . . . 4  |-  ( TC
`  a )  C_  U.
ran  ( U `  a )
27 unissb 4194 . . . . 5  |-  ( U. ran  ( U `  a
)  C_  ( TC `  a )  <->  A. b  e.  ran  ( U `  a ) b  C_  ( TC `  a ) )
28 fvelrnb 5821 . . . . . . 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 8713 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  C_  ( TC `  a )
3130a1i 11 . . . . . . . 8  |-  ( c  e.  om  ->  (
( U `  a
) `  c )  C_  ( TC `  a
) )
32 sseq1 3438 . . . . . . . 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 2868 . . . . . 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 2746 . . . 4  |-  U. ran  ( U `  a ) 
C_  ( TC `  a )
3726, 36eqssi 3433 . . 3  |-  ( TC
`  a )  = 
U. ran  ( U `  a )
385, 37vtoclg 3092 . 2  |-  ( A  e.  _V  ->  ( TC `  A )  = 
U. ran  ( U `  A ) )
39 rn0 5167 . . . . 5  |-  ran  (/)  =  (/)
4039unieqi 4172 . . . 4  |-  U. ran  (/)  =  U. (/)
41 uni0 4190 . . . 4  |-  U. (/)  =  (/)
4240, 41eqtr2i 2412 . . 3  |-  (/)  =  U. ran  (/)
43 fvprc 5768 . . 3  |-  ( -.  A  e.  _V  ->  ( TC `  A )  =  (/) )
44 fvprc 5768 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544rneqd 5143 . . . 4  |-  ( -.  A  e.  _V  ->  ran  ( U `  A
)  =  ran  (/) )
4645unieqd 4173 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  ( U `  A )  =  U. ran  (/) )
4742, 43, 463eqtr4a 2449 . 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 367    = wceq 1399    e. wcel 1826   E.wrex 2733   _Vcvv 3034    C_ wss 3389   (/)c0 3711   U.cuni 4163    |-> cmpt 4425   Tr wtr 4460   suc csuc 4794   ran crn 4914    |` cres 4915    Fn wfn 5491   ` cfv 5496   omcom 6599   reccrdg 6993   TCctc 8080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994  df-tc 8081
This theorem is referenced by:  hsmexlem5  8723
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