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Theorem itunitc1 8062
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 5541 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
21fveq1d 5543 . . . 4  |-  ( a  =  A  ->  (
( U `  a
) `  B )  =  ( ( U `
 A ) `  B ) )
3 fveq2 5541 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
42, 3sseq12d 3220 . . 3  |-  ( a  =  A  ->  (
( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  ( ( U `  A ) `  B )  C_  ( TC `  A ) ) )
5 fveq2 5541 . . . . . 6  |-  ( b  =  (/)  ->  ( ( U `  a ) `
 b )  =  ( ( U `  a ) `  (/) ) )
65sseq1d 3218 . . . . 5  |-  ( b  =  (/)  ->  ( ( ( U `  a
) `  b )  C_  ( TC `  a
)  <->  ( ( U `
 a ) `  (/) )  C_  ( TC `  a ) ) )
7 fveq2 5541 . . . . . 6  |-  ( b  =  c  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  c ) )
87sseq1d 3218 . . . . 5  |-  ( b  =  c  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) ) )
9 fveq2 5541 . . . . . 6  |-  ( b  =  suc  c  -> 
( ( U `  a ) `  b
)  =  ( ( U `  a ) `
 suc  c )
)
109sseq1d 3218 . . . . 5  |-  ( b  =  suc  c  -> 
( ( ( U `
 a ) `  b )  C_  ( TC `  a )  <->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) ) )
11 fveq2 5541 . . . . . 6  |-  ( b  =  B  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  B ) )
1211sseq1d 3218 . . . . 5  |-  ( b  =  B  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  B )  C_  ( TC `  a ) ) )
13 vex 2804 . . . . . 6  |-  a  e. 
_V
14 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
1514ituni0 8060 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
16 tcid 7440 . . . . . . 7  |-  ( a  e.  _V  ->  a  C_  ( TC `  a
) )
1715, 16eqsstrd 3225 . . . . . 6  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  C_  ( TC `  a ) )
1813, 17ax-mp 8 . . . . 5  |-  ( ( U `  a ) `
 (/) )  C_  ( TC `  a )
1914itunisuc 8061 . . . . . . 7  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
20 tctr 7441 . . . . . . . . . 10  |-  Tr  ( TC `  a )
21 pwtr 4242 . . . . . . . . . 10  |-  ( Tr  ( TC `  a
)  <->  Tr  ~P ( TC `  a ) )
2220, 21mpbi 199 . . . . . . . . 9  |-  Tr  ~P ( TC `  a )
23 trss 4138 . . . . . . . . 9  |-  ( Tr 
~P ( TC `  a )  ->  (
( ( U `  a ) `  c
)  e.  ~P ( TC `  a )  -> 
( ( U `  a ) `  c
)  C_  ~P ( TC `  a ) ) )
2422, 23ax-mp 8 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  ->  (
( U `  a
) `  c )  C_ 
~P ( TC `  a ) )
25 fvex 5555 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  e. 
_V
2625elpw 3644 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) )
27 sspwuni 4003 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  C_ 
~P ( TC `  a )  <->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2824, 26, 273imtr3i 256 . . . . . . 7  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2919, 28syl5eqss 3235 . . . . . 6  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) )
3029a1i 10 . . . . 5  |-  ( c  e.  om  ->  (
( ( U `  a ) `  c
)  C_  ( TC `  a )  ->  (
( U `  a
) `  suc  c ) 
C_  ( TC `  a ) ) )
316, 8, 10, 12, 18, 30finds 4698 . . . 4  |-  ( B  e.  om  ->  (
( U `  a
) `  B )  C_  ( TC `  a
) )
32 0ss 3496 . . . . 5  |-  (/)  C_  ( TC `  a )
3314itunifn 8059 . . . . . . . . 9  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
34 fndm 5359 . . . . . . . . 9  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
3513, 33, 34mp2b 9 . . . . . . . 8  |-  dom  ( U `  a )  =  om
3635eleq2i 2360 . . . . . . 7  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
37 ndmfv 5568 . . . . . . 7  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
3836, 37sylnbir 298 . . . . . 6  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
3938sseq1d 3218 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  (/)  C_  ( TC `  a ) ) )
4032, 39mpbiri 224 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  C_  ( TC `  a
) )
4131, 40pm2.61i 156 . . 3  |-  ( ( U `  a ) `
 B )  C_  ( TC `  a )
424, 41vtoclg 2856 . 2  |-  ( A  e.  _V  ->  (
( U `  A
) `  B )  C_  ( TC `  A
) )
43 0ss 3496 . . 3  |-  (/)  C_  ( TC `  A )
44 fvprc 5535 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4544fveq1d 5543 . . . . 5  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  ( (/) `  B
) )
46 fv01 5575 . . . . 5  |-  ( (/) `  B )  =  (/)
4745, 46syl6eq 2344 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  (/) )
4847sseq1d 3218 . . 3  |-  ( -.  A  e.  _V  ->  ( ( ( U `  A ) `  B
)  C_  ( TC `  A )  <->  (/)  C_  ( TC `  A ) ) )
4943, 48mpbiri 224 . 2  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  C_  ( TC `  A
) )
5042, 49pm2.61i 156 1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843    e. cmpt 4093   Tr wtr 4129   suc csuc 4410   omcom 4672   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271   reccrdg 6438   TCctc 7437
This theorem is referenced by:  itunitc  8063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-tc 7438
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