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Theorem ituniiun 8798
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 5864 . . . 4  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
21fveq1d 5866 . . 3  |-  ( b  =  A  ->  (
( U `  b
) `  suc  B )  =  ( ( U `
 A ) `  suc  B ) )
3 iuneq1 4339 . . 3  |-  ( b  =  A  ->  U_ a  e.  b  ( ( U `  a ) `  B )  =  U_ a  e.  A  (
( U `  a
) `  B )
)
42, 3eqeq12d 2489 . 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 4943 . . . . . 6  |-  ( d  =  (/)  ->  suc  d  =  suc  (/) )
65fveq2d 5868 . . . . 5  |-  ( d  =  (/)  ->  ( ( U `  b ) `
 suc  d )  =  ( ( U `
 b ) `  suc  (/) ) )
7 fveq2 5864 . . . . . 6  |-  ( d  =  (/)  ->  ( ( U `  a ) `
 d )  =  ( ( U `  a ) `  (/) ) )
87iuneq2d 4352 . . . . 5  |-  ( d  =  (/)  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) )
96, 8eqeq12d 2489 . . . 4  |-  ( d  =  (/)  ->  ( ( ( U `  b
) `  suc  d )  =  U_ a  e.  b  ( ( U `
 a ) `  d )  <->  ( ( U `  b ) `  suc  (/) )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) ) )
10 suceq 4943 . . . . . 6  |-  ( d  =  c  ->  suc  d  =  suc  c )
1110fveq2d 5868 . . . . 5  |-  ( d  =  c  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  c ) )
12 fveq2 5864 . . . . . 6  |-  ( d  =  c  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  c ) )
1312iuneq2d 4352 . . . . 5  |-  ( d  =  c  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  c )
)
1411, 13eqeq12d 2489 . . . 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 4943 . . . . . 6  |-  ( d  =  suc  c  ->  suc  d  =  suc  suc  c )
1615fveq2d 5868 . . . . 5  |-  ( d  =  suc  c  -> 
( ( U `  b ) `  suc  d )  =  ( ( U `  b
) `  suc  suc  c
) )
17 fveq2 5864 . . . . . 6  |-  ( d  =  suc  c  -> 
( ( U `  a ) `  d
)  =  ( ( U `  a ) `
 suc  c )
)
1817iuneq2d 4352 . . . . 5  |-  ( d  =  suc  c  ->  U_ a  e.  b 
( ( U `  a ) `  d
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) )
1916, 18eqeq12d 2489 . . . 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 4943 . . . . . 6  |-  ( d  =  B  ->  suc  d  =  suc  B )
2120fveq2d 5868 . . . . 5  |-  ( d  =  B  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  B ) )
22 fveq2 5864 . . . . . 6  |-  ( d  =  B  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  B ) )
2322iuneq2d 4352 . . . . 5  |-  ( d  =  B  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  B )
)
2421, 23eqeq12d 2489 . . . 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 4378 . . . . 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 8795 . . . . . 6  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. ( ( U `
 b ) `  (/) )
28 vex 3116 . . . . . . . 8  |-  b  e. 
_V
2926ituni0 8794 . . . . . . . 8  |-  ( b  e.  _V  ->  (
( U `  b
) `  (/) )  =  b )
3028, 29ax-mp 5 . . . . . . 7  |-  ( ( U `  b ) `
 (/) )  =  b
3130unieqi 4254 . . . . . 6  |-  U. (
( U `  b
) `  (/) )  = 
U. b
3227, 31eqtri 2496 . . . . 5  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. b
3326ituni0 8794 . . . . . 6  |-  ( a  e.  b  ->  (
( U `  a
) `  (/) )  =  a )
3433iuneq2i 4344 . . . . 5  |-  U_ a  e.  b  ( ( U `  a ) `  (/) )  =  U_ a  e.  b  a
3525, 32, 343eqtr4i 2506 . . . 4  |-  ( ( U `  b ) `
 suc  (/) )  = 
U_ a  e.  b  ( ( U `  a ) `  (/) )
3626itunisuc 8795 . . . . . 6  |-  ( ( U `  b ) `
 suc  suc  c )  =  U. ( ( U `  b ) `
 suc  c )
37 unieq 4253 . . . . . . 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 8795 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
3938a1i 11 . . . . . . . . 9  |-  ( a  e.  b  ->  (
( U `  a
) `  suc  c )  =  U. ( ( U `  a ) `
 c ) )
4039iuneq2i 4344 . . . . . . . 8  |-  U_ a  e.  b  ( ( U `  a ) `  suc  c )  = 
U_ a  e.  b 
U. ( ( U `
 a ) `  c )
41 iuncom4 4333 . . . . . . . 8  |-  U_ a  e.  b  U. (
( U `  a
) `  c )  =  U. U_ a  e.  b  ( ( U `
 a ) `  c )
4240, 41eqtr2i 2497 . . . . . . 7  |-  U. U_ a  e.  b  (
( U `  a
) `  c )  =  U_ a  e.  b  ( ( U `  a ) `  suc  c )
4337, 42syl6eq 2524 . . . . . 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 2520 . . . . 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 6704 . . 3  |-  ( B  e.  om  ->  (
( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
47 iun0 4381 . . . . 5  |-  U_ a  e.  b  (/)  =  (/)
4847eqcomi 2480 . . . 4  |-  (/)  =  U_ a  e.  b  (/)
49 peano2b 6694 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
5026itunifn 8793 . . . . . . . 8  |-  ( b  e.  _V  ->  ( U `  b )  Fn  om )
51 fndm 5678 . . . . . . . 8  |-  ( ( U `  b )  Fn  om  ->  dom  ( U `  b )  =  om )
5228, 50, 51mp2b 10 . . . . . . 7  |-  dom  ( U `  b )  =  om
5352eleq2i 2545 . . . . . 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 5888 . . . . 5  |-  ( -. 
suc  B  e.  dom  ( U `  b )  ->  ( ( U `
 b ) `  suc  B )  =  (/) )
5654, 55sylnbi 306 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  (/) )
57 vex 3116 . . . . . . . 8  |-  a  e. 
_V
5826itunifn 8793 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
59 fndm 5678 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
6057, 58, 59mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
6160eleq2i 2545 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
62 ndmfv 5888 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
6361, 62sylnbir 307 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
6463iuneq2d 4352 . . . 4  |-  ( -.  B  e.  om  ->  U_ a  e.  b  ( ( U `  a
) `  B )  =  U_ a  e.  b  (/) )
6548, 56, 643eqtr4a 2534 . . 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 3171 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 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   U.cuni 4245   U_ciun 4325    |-> cmpt 4505   suc csuc 4880   dom cdm 4999    |` cres 5001    Fn wfn 5581   ` cfv 5586   omcom 6678   reccrdg 7072
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 6574  ax-inf2 8054
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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073
This theorem is referenced by:  hsmexlem4  8805
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