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Theorem itunitc1 8848
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 5881 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
21fveq1d 5883 . . . 4  |-  ( a  =  A  ->  (
( U `  a
) `  B )  =  ( ( U `
 A ) `  B ) )
3 fveq2 5881 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
42, 3sseq12d 3499 . . 3  |-  ( a  =  A  ->  (
( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  ( ( U `  A ) `  B )  C_  ( TC `  A ) ) )
5 fveq2 5881 . . . . . 6  |-  ( b  =  (/)  ->  ( ( U `  a ) `
 b )  =  ( ( U `  a ) `  (/) ) )
65sseq1d 3497 . . . . 5  |-  ( b  =  (/)  ->  ( ( ( U `  a
) `  b )  C_  ( TC `  a
)  <->  ( ( U `
 a ) `  (/) )  C_  ( TC `  a ) ) )
7 fveq2 5881 . . . . . 6  |-  ( b  =  c  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  c ) )
87sseq1d 3497 . . . . 5  |-  ( b  =  c  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) ) )
9 fveq2 5881 . . . . . 6  |-  ( b  =  suc  c  -> 
( ( U `  a ) `  b
)  =  ( ( U `  a ) `
 suc  c )
)
109sseq1d 3497 . . . . 5  |-  ( b  =  suc  c  -> 
( ( ( U `
 a ) `  b )  C_  ( TC `  a )  <->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) ) )
11 fveq2 5881 . . . . . 6  |-  ( b  =  B  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  B ) )
1211sseq1d 3497 . . . . 5  |-  ( b  =  B  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  B )  C_  ( TC `  a ) ) )
13 vex 3090 . . . . . 6  |-  a  e. 
_V
14 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
1514ituni0 8846 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
16 tcid 8222 . . . . . . 7  |-  ( a  e.  _V  ->  a  C_  ( TC `  a
) )
1715, 16eqsstrd 3504 . . . . . 6  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  C_  ( TC `  a ) )
1813, 17ax-mp 5 . . . . 5  |-  ( ( U `  a ) `
 (/) )  C_  ( TC `  a )
1914itunisuc 8847 . . . . . . 7  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
20 tctr 8223 . . . . . . . . . 10  |-  Tr  ( TC `  a )
21 pwtr 4675 . . . . . . . . . 10  |-  ( Tr  ( TC `  a
)  <->  Tr  ~P ( TC `  a ) )
2220, 21mpbi 211 . . . . . . . . 9  |-  Tr  ~P ( TC `  a )
23 trss 4529 . . . . . . . . 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 5891 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  e. 
_V
2625elpw 3991 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) )
27 sspwuni 4391 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  C_ 
~P ( TC `  a )  <->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2824, 26, 273imtr3i 268 . . . . . . 7  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2919, 28syl5eqss 3514 . . . . . 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 6733 . . . 4  |-  ( B  e.  om  ->  (
( U `  a
) `  B )  C_  ( TC `  a
) )
3214itunifn 8845 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
33 fndm 5693 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
3413, 32, 33mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
3534eleq2i 2507 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
36 ndmfv 5905 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
3735, 36sylnbir 308 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
38 0ss 3797 . . . . 5  |-  (/)  C_  ( TC `  a )
3937, 38syl6eqss 3520 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  C_  ( TC `  a
) )
4031, 39pm2.61i 167 . . 3  |-  ( ( U `  a ) `
 B )  C_  ( TC `  a )
414, 40vtoclg 3145 . 2  |-  ( A  e.  _V  ->  (
( U `  A
) `  B )  C_  ( TC `  A
) )
42 fvprc 5875 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4342fveq1d 5883 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  ( (/) `  B
) )
44 0fv 5914 . . . 4  |-  ( (/) `  B )  =  (/)
4543, 44syl6eq 2486 . . 3  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  (/) )
46 0ss 3797 . . 3  |-  (/)  C_  ( TC `  A )
4745, 46syl6eqss 3520 . 2  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  C_  ( TC `  A
) )
4841, 47pm2.61i 167 1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   U.cuni 4222    |-> cmpt 4484   Tr wtr 4520   dom cdm 4854    |` cres 4856   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6706   reccrdg 7135   TCctc 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-tc 8220
This theorem is referenced by:  itunitc  8849
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