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Theorem itunitc1 8800
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 5866 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
21fveq1d 5868 . . . 4  |-  ( a  =  A  ->  (
( U `  a
) `  B )  =  ( ( U `
 A ) `  B ) )
3 fveq2 5866 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
42, 3sseq12d 3533 . . 3  |-  ( a  =  A  ->  (
( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  ( ( U `  A ) `  B )  C_  ( TC `  A ) ) )
5 fveq2 5866 . . . . . 6  |-  ( b  =  (/)  ->  ( ( U `  a ) `
 b )  =  ( ( U `  a ) `  (/) ) )
65sseq1d 3531 . . . . 5  |-  ( b  =  (/)  ->  ( ( ( U `  a
) `  b )  C_  ( TC `  a
)  <->  ( ( U `
 a ) `  (/) )  C_  ( TC `  a ) ) )
7 fveq2 5866 . . . . . 6  |-  ( b  =  c  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  c ) )
87sseq1d 3531 . . . . 5  |-  ( b  =  c  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) ) )
9 fveq2 5866 . . . . . 6  |-  ( b  =  suc  c  -> 
( ( U `  a ) `  b
)  =  ( ( U `  a ) `
 suc  c )
)
109sseq1d 3531 . . . . 5  |-  ( b  =  suc  c  -> 
( ( ( U `
 a ) `  b )  C_  ( TC `  a )  <->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) ) )
11 fveq2 5866 . . . . . 6  |-  ( b  =  B  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  B ) )
1211sseq1d 3531 . . . . 5  |-  ( b  =  B  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  B )  C_  ( TC `  a ) ) )
13 vex 3116 . . . . . 6  |-  a  e. 
_V
14 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
1514ituni0 8798 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
16 tcid 8170 . . . . . . 7  |-  ( a  e.  _V  ->  a  C_  ( TC `  a
) )
1715, 16eqsstrd 3538 . . . . . 6  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  C_  ( TC `  a ) )
1813, 17ax-mp 5 . . . . 5  |-  ( ( U `  a ) `
 (/) )  C_  ( TC `  a )
1914itunisuc 8799 . . . . . . 7  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
20 tctr 8171 . . . . . . . . . 10  |-  Tr  ( TC `  a )
21 pwtr 4700 . . . . . . . . . 10  |-  ( Tr  ( TC `  a
)  <->  Tr  ~P ( TC `  a ) )
2220, 21mpbi 208 . . . . . . . . 9  |-  Tr  ~P ( TC `  a )
23 trss 4549 . . . . . . . . 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 5876 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  e. 
_V
2625elpw 4016 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) )
27 sspwuni 4411 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  C_ 
~P ( TC `  a )  <->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2824, 26, 273imtr3i 265 . . . . . . 7  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2919, 28syl5eqss 3548 . . . . . 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 6710 . . . 4  |-  ( B  e.  om  ->  (
( U `  a
) `  B )  C_  ( TC `  a
) )
3214itunifn 8797 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
33 fndm 5680 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
3413, 32, 33mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
3534eleq2i 2545 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
36 ndmfv 5890 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
3735, 36sylnbir 307 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
38 0ss 3814 . . . . 5  |-  (/)  C_  ( TC `  a )
3937, 38syl6eqss 3554 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  C_  ( TC `  a
) )
4031, 39pm2.61i 164 . . 3  |-  ( ( U `  a ) `
 B )  C_  ( TC `  a )
414, 40vtoclg 3171 . 2  |-  ( A  e.  _V  ->  (
( U `  A
) `  B )  C_  ( TC `  A
) )
42 fvprc 5860 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4342fveq1d 5868 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  ( (/) `  B
) )
44 0fv 5899 . . . 4  |-  ( (/) `  B )  =  (/)
4543, 44syl6eq 2524 . . 3  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  (/) )
46 0ss 3814 . . 3  |-  (/)  C_  ( TC `  A )
4745, 46syl6eqss 3554 . 2  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  C_  ( TC `  A
) )
4841, 47pm2.61i 164 1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245    |-> cmpt 4505   Tr wtr 4540   suc csuc 4880   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588   omcom 6684   reccrdg 7075   TCctc 8167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-recs 7042  df-rdg 7076  df-tc 8168
This theorem is referenced by:  itunitc  8801
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