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Theorem ituniiun 8694
Description: Unwrap an iterated union from the "other end". (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
ituniiun  |-  ( A  e.  V  ->  (
( U `  A
) `  suc  B )  =  U_ a  e.  A  ( ( U `
 a ) `  B ) )
Distinct variable groups:    x, A, y, a    x, B, y, a    U, a
Allowed substitution hints:    U( x, y)    V( x, y, a)

Proof of Theorem ituniiun
Dummy variables  b 
c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5791 . . . 4  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
21fveq1d 5793 . . 3  |-  ( b  =  A  ->  (
( U `  b
) `  suc  B )  =  ( ( U `
 A ) `  suc  B ) )
3 iuneq1 4284 . . 3  |-  ( b  =  A  ->  U_ a  e.  b  ( ( U `  a ) `  B )  =  U_ a  e.  A  (
( U `  a
) `  B )
)
42, 3eqeq12d 2473 . 2  |-  ( b  =  A  ->  (
( ( U `  b ) `  suc  B )  =  U_ a  e.  b  ( ( U `  a ) `  B )  <->  ( ( U `  A ) `  suc  B )  = 
U_ a  e.  A  ( ( U `  a ) `  B
) ) )
5 suceq 4884 . . . . . 6  |-  ( d  =  (/)  ->  suc  d  =  suc  (/) )
65fveq2d 5795 . . . . 5  |-  ( d  =  (/)  ->  ( ( U `  b ) `
 suc  d )  =  ( ( U `
 b ) `  suc  (/) ) )
7 fveq2 5791 . . . . . 6  |-  ( d  =  (/)  ->  ( ( U `  a ) `
 d )  =  ( ( U `  a ) `  (/) ) )
87iuneq2d 4297 . . . . 5  |-  ( d  =  (/)  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) )
96, 8eqeq12d 2473 . . . 4  |-  ( d  =  (/)  ->  ( ( ( U `  b
) `  suc  d )  =  U_ a  e.  b  ( ( U `
 a ) `  d )  <->  ( ( U `  b ) `  suc  (/) )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) ) )
10 suceq 4884 . . . . . 6  |-  ( d  =  c  ->  suc  d  =  suc  c )
1110fveq2d 5795 . . . . 5  |-  ( d  =  c  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  c ) )
12 fveq2 5791 . . . . . 6  |-  ( d  =  c  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  c ) )
1312iuneq2d 4297 . . . . 5  |-  ( d  =  c  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  c )
)
1411, 13eqeq12d 2473 . . . 4  |-  ( d  =  c  ->  (
( ( U `  b ) `  suc  d )  =  U_ a  e.  b  (
( U `  a
) `  d )  <->  ( ( U `  b
) `  suc  c )  =  U_ a  e.  b  ( ( U `
 a ) `  c ) ) )
15 suceq 4884 . . . . . 6  |-  ( d  =  suc  c  ->  suc  d  =  suc  suc  c )
1615fveq2d 5795 . . . . 5  |-  ( d  =  suc  c  -> 
( ( U `  b ) `  suc  d )  =  ( ( U `  b
) `  suc  suc  c
) )
17 fveq2 5791 . . . . . 6  |-  ( d  =  suc  c  -> 
( ( U `  a ) `  d
)  =  ( ( U `  a ) `
 suc  c )
)
1817iuneq2d 4297 . . . . 5  |-  ( d  =  suc  c  ->  U_ a  e.  b 
( ( U `  a ) `  d
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) )
1916, 18eqeq12d 2473 . . . 4  |-  ( d  =  suc  c  -> 
( ( ( U `
 b ) `  suc  d )  =  U_ a  e.  b  (
( U `  a
) `  d )  <->  ( ( U `  b
) `  suc  suc  c
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) ) )
20 suceq 4884 . . . . . 6  |-  ( d  =  B  ->  suc  d  =  suc  B )
2120fveq2d 5795 . . . . 5  |-  ( d  =  B  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  B ) )
22 fveq2 5791 . . . . . 6  |-  ( d  =  B  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  B ) )
2322iuneq2d 4297 . . . . 5  |-  ( d  =  B  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  B )
)
2421, 23eqeq12d 2473 . . . 4  |-  ( d  =  B  ->  (
( ( U `  b ) `  suc  d )  =  U_ a  e.  b  (
( U `  a
) `  d )  <->  ( ( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) ) )
25 uniiun 4323 . . . . 5  |-  U. b  =  U_ a  e.  b  a
26 ituni.u . . . . . . 7  |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
) ,  x )  |`  om ) )
2726itunisuc 8691 . . . . . 6  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. ( ( U `
 b ) `  (/) )
28 vex 3073 . . . . . . . 8  |-  b  e. 
_V
2926ituni0 8690 . . . . . . . 8  |-  ( b  e.  _V  ->  (
( U `  b
) `  (/) )  =  b )
3028, 29ax-mp 5 . . . . . . 7  |-  ( ( U `  b ) `
 (/) )  =  b
3130unieqi 4200 . . . . . 6  |-  U. (
( U `  b
) `  (/) )  = 
U. b
3227, 31eqtri 2480 . . . . 5  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. b
3326ituni0 8690 . . . . . 6  |-  ( a  e.  b  ->  (
( U `  a
) `  (/) )  =  a )
3433iuneq2i 4289 . . . . 5  |-  U_ a  e.  b  ( ( U `  a ) `  (/) )  =  U_ a  e.  b  a
3525, 32, 343eqtr4i 2490 . . . 4  |-  ( ( U `  b ) `
 suc  (/) )  = 
U_ a  e.  b  ( ( U `  a ) `  (/) )
3626itunisuc 8691 . . . . . 6  |-  ( ( U `  b ) `
 suc  suc  c )  =  U. ( ( U `  b ) `
 suc  c )
37 unieq 4199 . . . . . . 7  |-  ( ( ( U `  b
) `  suc  c )  =  U_ a  e.  b  ( ( U `
 a ) `  c )  ->  U. (
( U `  b
) `  suc  c )  =  U. U_ a  e.  b  ( ( U `  a ) `  c ) )
3826itunisuc 8691 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
3938a1i 11 . . . . . . . . 9  |-  ( a  e.  b  ->  (
( U `  a
) `  suc  c )  =  U. ( ( U `  a ) `
 c ) )
4039iuneq2i 4289 . . . . . . . 8  |-  U_ a  e.  b  ( ( U `  a ) `  suc  c )  = 
U_ a  e.  b 
U. ( ( U `
 a ) `  c )
41 iuncom4 4278 . . . . . . . 8  |-  U_ a  e.  b  U. (
( U `  a
) `  c )  =  U. U_ a  e.  b  ( ( U `
 a ) `  c )
4240, 41eqtr2i 2481 . . . . . . 7  |-  U. U_ a  e.  b  (
( U `  a
) `  c )  =  U_ a  e.  b  ( ( U `  a ) `  suc  c )
4337, 42syl6eq 2508 . . . . . 6  |-  ( ( ( U `  b
) `  suc  c )  =  U_ a  e.  b  ( ( U `
 a ) `  c )  ->  U. (
( U `  b
) `  suc  c )  =  U_ a  e.  b  ( ( U `
 a ) `  suc  c ) )
4436, 43syl5eq 2504 . . . . 5  |-  ( ( ( U `  b
) `  suc  c )  =  U_ a  e.  b  ( ( U `
 a ) `  c )  ->  (
( U `  b
) `  suc  suc  c
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) )
4544a1i 11 . . . 4  |-  ( c  e.  om  ->  (
( ( U `  b ) `  suc  c )  =  U_ a  e.  b  (
( U `  a
) `  c )  ->  ( ( U `  b ) `  suc  suc  c )  =  U_ a  e.  b  (
( U `  a
) `  suc  c ) ) )
469, 14, 19, 24, 35, 45finds 6604 . . 3  |-  ( B  e.  om  ->  (
( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
47 iun0 4326 . . . . 5  |-  U_ a  e.  b  (/)  =  (/)
4847eqcomi 2464 . . . 4  |-  (/)  =  U_ a  e.  b  (/)
49 peano2b 6594 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
5026itunifn 8689 . . . . . . . 8  |-  ( b  e.  _V  ->  ( U `  b )  Fn  om )
51 fndm 5610 . . . . . . . 8  |-  ( ( U `  b )  Fn  om  ->  dom  ( U `  b )  =  om )
5228, 50, 51mp2b 10 . . . . . . 7  |-  dom  ( U `  b )  =  om
5352eleq2i 2529 . . . . . 6  |-  ( suc 
B  e.  dom  ( U `  b )  <->  suc 
B  e.  om )
5449, 53bitr4i 252 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
dom  ( U `  b ) )
55 ndmfv 5815 . . . . 5  |-  ( -. 
suc  B  e.  dom  ( U `  b )  ->  ( ( U `
 b ) `  suc  B )  =  (/) )
5654, 55sylnbi 306 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  (/) )
57 vex 3073 . . . . . . . 8  |-  a  e. 
_V
5826itunifn 8689 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
59 fndm 5610 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
6057, 58, 59mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
6160eleq2i 2529 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
62 ndmfv 5815 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
6361, 62sylnbir 307 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
6463iuneq2d 4297 . . . 4  |-  ( -.  B  e.  om  ->  U_ a  e.  b  ( ( U `  a
) `  B )  =  U_ a  e.  b  (/) )
6548, 56, 643eqtr4a 2518 . . 3  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
6646, 65pm2.61i 164 . 2  |-  ( ( U `  b ) `
 suc  B )  =  U_ a  e.  b  ( ( U `  a ) `  B
)
674, 66vtoclg 3128 1  |-  ( A  e.  V  ->  (
( U `  A
) `  suc  B )  =  U_ a  e.  A  ( ( U `
 a ) `  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737   U.cuni 4191   U_ciun 4271    |-> cmpt 4450   suc csuc 4821   dom cdm 4940    |` cres 4942    Fn wfn 5513   ` cfv 5518   omcom 6578   reccrdg 6967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-recs 6934  df-rdg 6968
This theorem is referenced by:  hsmexlem4  8701
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