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Theorem itunitc1 8868
Description: Each union iterate is a member of the transitive closure. (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
itunitc1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Distinct variable groups:    x, A, y    x, B, y
Allowed substitution hints:    U( x, y)

Proof of Theorem itunitc1
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5879 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
21fveq1d 5881 . . . 4  |-  ( a  =  A  ->  (
( U `  a
) `  B )  =  ( ( U `
 A ) `  B ) )
3 fveq2 5879 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
42, 3sseq12d 3447 . . 3  |-  ( a  =  A  ->  (
( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  ( ( U `  A ) `  B )  C_  ( TC `  A ) ) )
5 fveq2 5879 . . . . . 6  |-  ( b  =  (/)  ->  ( ( U `  a ) `
 b )  =  ( ( U `  a ) `  (/) ) )
65sseq1d 3445 . . . . 5  |-  ( b  =  (/)  ->  ( ( ( U `  a
) `  b )  C_  ( TC `  a
)  <->  ( ( U `
 a ) `  (/) )  C_  ( TC `  a ) ) )
7 fveq2 5879 . . . . . 6  |-  ( b  =  c  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  c ) )
87sseq1d 3445 . . . . 5  |-  ( b  =  c  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) ) )
9 fveq2 5879 . . . . . 6  |-  ( b  =  suc  c  -> 
( ( U `  a ) `  b
)  =  ( ( U `  a ) `
 suc  c )
)
109sseq1d 3445 . . . . 5  |-  ( b  =  suc  c  -> 
( ( ( U `
 a ) `  b )  C_  ( TC `  a )  <->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) ) )
11 fveq2 5879 . . . . . 6  |-  ( b  =  B  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  B ) )
1211sseq1d 3445 . . . . 5  |-  ( b  =  B  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  B )  C_  ( TC `  a ) ) )
13 vex 3034 . . . . . 6  |-  a  e. 
_V
14 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
1514ituni0 8866 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
16 tcid 8241 . . . . . . 7  |-  ( a  e.  _V  ->  a  C_  ( TC `  a
) )
1715, 16eqsstrd 3452 . . . . . 6  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  C_  ( TC `  a ) )
1813, 17ax-mp 5 . . . . 5  |-  ( ( U `  a ) `
 (/) )  C_  ( TC `  a )
1914itunisuc 8867 . . . . . . 7  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
20 tctr 8242 . . . . . . . . . 10  |-  Tr  ( TC `  a )
21 pwtr 4653 . . . . . . . . . 10  |-  ( Tr  ( TC `  a
)  <->  Tr  ~P ( TC `  a ) )
2220, 21mpbi 213 . . . . . . . . 9  |-  Tr  ~P ( TC `  a )
23 trss 4499 . . . . . . . . 9  |-  ( Tr 
~P ( TC `  a )  ->  (
( ( U `  a ) `  c
)  e.  ~P ( TC `  a )  -> 
( ( U `  a ) `  c
)  C_  ~P ( TC `  a ) ) )
2422, 23ax-mp 5 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  ->  (
( U `  a
) `  c )  C_ 
~P ( TC `  a ) )
25 fvex 5889 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  e. 
_V
2625elpw 3948 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) )
27 sspwuni 4360 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  C_ 
~P ( TC `  a )  <->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2824, 26, 273imtr3i 273 . . . . . . 7  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2919, 28syl5eqss 3462 . . . . . 6  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) )
3029a1i 11 . . . . 5  |-  ( c  e.  om  ->  (
( ( U `  a ) `  c
)  C_  ( TC `  a )  ->  (
( U `  a
) `  suc  c ) 
C_  ( TC `  a ) ) )
316, 8, 10, 12, 18, 30finds 6738 . . . 4  |-  ( B  e.  om  ->  (
( U `  a
) `  B )  C_  ( TC `  a
) )
3214itunifn 8865 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
33 fndm 5685 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
3413, 32, 33mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
3534eleq2i 2541 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
36 ndmfv 5903 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
3735, 36sylnbir 314 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
38 0ss 3766 . . . . 5  |-  (/)  C_  ( TC `  a )
3937, 38syl6eqss 3468 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  C_  ( TC `  a
) )
4031, 39pm2.61i 169 . . 3  |-  ( ( U `  a ) `
 B )  C_  ( TC `  a )
414, 40vtoclg 3093 . 2  |-  ( A  e.  _V  ->  (
( U `  A
) `  B )  C_  ( TC `  A
) )
42 fvprc 5873 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4342fveq1d 5881 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  ( (/) `  B
) )
44 0fv 5912 . . . 4  |-  ( (/) `  B )  =  (/)
4543, 44syl6eq 2521 . . 3  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  (/) )
46 0ss 3766 . . 3  |-  (/)  C_  ( TC `  A )
4745, 46syl6eqss 3468 . 2  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  C_  ( TC `  A
) )
4841, 47pm2.61i 169 1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190    |-> cmpt 4454   Tr wtr 4490   dom cdm 4839    |` cres 4841   suc csuc 5432    Fn wfn 5584   ` cfv 5589   omcom 6711   reccrdg 7145   TCctc 8238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-tc 8239
This theorem is referenced by:  itunitc  8869
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