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Theorem itunitc1 8302
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 5730 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
21fveq1d 5732 . . . 4  |-  ( a  =  A  ->  (
( U `  a
) `  B )  =  ( ( U `
 A ) `  B ) )
3 fveq2 5730 . . . 4  |-  ( a  =  A  ->  ( TC `  a )  =  ( TC `  A
) )
42, 3sseq12d 3379 . . 3  |-  ( a  =  A  ->  (
( ( U `  a ) `  B
)  C_  ( TC `  a )  <->  ( ( U `  A ) `  B )  C_  ( TC `  A ) ) )
5 fveq2 5730 . . . . . 6  |-  ( b  =  (/)  ->  ( ( U `  a ) `
 b )  =  ( ( U `  a ) `  (/) ) )
65sseq1d 3377 . . . . 5  |-  ( b  =  (/)  ->  ( ( ( U `  a
) `  b )  C_  ( TC `  a
)  <->  ( ( U `
 a ) `  (/) )  C_  ( TC `  a ) ) )
7 fveq2 5730 . . . . . 6  |-  ( b  =  c  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  c ) )
87sseq1d 3377 . . . . 5  |-  ( b  =  c  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) ) )
9 fveq2 5730 . . . . . 6  |-  ( b  =  suc  c  -> 
( ( U `  a ) `  b
)  =  ( ( U `  a ) `
 suc  c )
)
109sseq1d 3377 . . . . 5  |-  ( b  =  suc  c  -> 
( ( ( U `
 a ) `  b )  C_  ( TC `  a )  <->  ( ( U `  a ) `  suc  c )  C_  ( TC `  a ) ) )
11 fveq2 5730 . . . . . 6  |-  ( b  =  B  ->  (
( U `  a
) `  b )  =  ( ( U `
 a ) `  B ) )
1211sseq1d 3377 . . . . 5  |-  ( b  =  B  ->  (
( ( U `  a ) `  b
)  C_  ( TC `  a )  <->  ( ( U `  a ) `  B )  C_  ( TC `  a ) ) )
13 vex 2961 . . . . . 6  |-  a  e. 
_V
14 ituni.u . . . . . . . 8  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
1514ituni0 8300 . . . . . . 7  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  =  a )
16 tcid 7680 . . . . . . 7  |-  ( a  e.  _V  ->  a  C_  ( TC `  a
) )
1715, 16eqsstrd 3384 . . . . . 6  |-  ( a  e.  _V  ->  (
( U `  a
) `  (/) )  C_  ( TC `  a ) )
1813, 17ax-mp 8 . . . . 5  |-  ( ( U `  a ) `
 (/) )  C_  ( TC `  a )
1914itunisuc 8301 . . . . . . 7  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
20 tctr 7681 . . . . . . . . . 10  |-  Tr  ( TC `  a )
21 pwtr 4418 . . . . . . . . . 10  |-  ( Tr  ( TC `  a
)  <->  Tr  ~P ( TC `  a ) )
2220, 21mpbi 201 . . . . . . . . 9  |-  Tr  ~P ( TC `  a )
23 trss 4313 . . . . . . . . 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 5744 . . . . . . . . 9  |-  ( ( U `  a ) `
 c )  e. 
_V
2625elpw 3807 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  e.  ~P ( TC `  a )  <->  ( ( U `  a ) `  c )  C_  ( TC `  a ) )
27 sspwuni 4178 . . . . . . . 8  |-  ( ( ( U `  a
) `  c )  C_ 
~P ( TC `  a )  <->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2824, 26, 273imtr3i 258 . . . . . . 7  |-  ( ( ( U `  a
) `  c )  C_  ( TC `  a
)  ->  U. (
( U `  a
) `  c )  C_  ( TC `  a
) )
2919, 28syl5eqss 3394 . . . . . 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 4873 . . . 4  |-  ( B  e.  om  ->  (
( U `  a
) `  B )  C_  ( TC `  a
) )
3214itunifn 8299 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
33 fndm 5546 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
3413, 32, 33mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
3534eleq2i 2502 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
36 ndmfv 5757 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
3735, 36sylnbir 300 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
38 0ss 3658 . . . . 5  |-  (/)  C_  ( TC `  a )
3937, 38syl6eqss 3400 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  C_  ( TC `  a
) )
4031, 39pm2.61i 159 . . 3  |-  ( ( U `  a ) `
 B )  C_  ( TC `  a )
414, 40vtoclg 3013 . 2  |-  ( A  e.  _V  ->  (
( U `  A
) `  B )  C_  ( TC `  A
) )
42 fvprc 5724 . . . . 5  |-  ( -.  A  e.  _V  ->  ( U `  A )  =  (/) )
4342fveq1d 5732 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  ( (/) `  B
) )
44 fv01 5765 . . . 4  |-  ( (/) `  B )  =  (/)
4543, 44syl6eq 2486 . . 3  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  =  (/) )
46 0ss 3658 . . 3  |-  (/)  C_  ( TC `  A )
4745, 46syl6eqss 3400 . 2  |-  ( -.  A  e.  _V  ->  ( ( U `  A
) `  B )  C_  ( TC `  A
) )
4841, 47pm2.61i 159 1  |-  ( ( U `  A ) `
 B )  C_  ( TC `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017    e. cmpt 4268   Tr wtr 4304   suc csuc 4585   omcom 4847   dom cdm 4880    |` cres 4882    Fn wfn 5451   ` cfv 5456   reccrdg 6669   TCctc 7677
This theorem is referenced by:  itunitc  8303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-tc 7678
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