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Theorem ituniiun 8195
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 5632 . . . 4  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
21fveq1d 5634 . . 3  |-  ( b  =  A  ->  (
( U `  b
) `  suc  B )  =  ( ( U `
 A ) `  suc  B ) )
3 iuneq1 4020 . . 3  |-  ( b  =  A  ->  U_ a  e.  b  ( ( U `  a ) `  B )  =  U_ a  e.  A  (
( U `  a
) `  B )
)
42, 3eqeq12d 2380 . 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 4560 . . . . . 6  |-  ( d  =  (/)  ->  suc  d  =  suc  (/) )
65fveq2d 5636 . . . . 5  |-  ( d  =  (/)  ->  ( ( U `  b ) `
 suc  d )  =  ( ( U `
 b ) `  suc  (/) ) )
7 fveq2 5632 . . . . . 6  |-  ( d  =  (/)  ->  ( ( U `  a ) `
 d )  =  ( ( U `  a ) `  (/) ) )
87iuneq2d 4032 . . . . 5  |-  ( d  =  (/)  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) )
96, 8eqeq12d 2380 . . . 4  |-  ( d  =  (/)  ->  ( ( ( U `  b
) `  suc  d )  =  U_ a  e.  b  ( ( U `
 a ) `  d )  <->  ( ( U `  b ) `  suc  (/) )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) ) )
10 suceq 4560 . . . . . 6  |-  ( d  =  c  ->  suc  d  =  suc  c )
1110fveq2d 5636 . . . . 5  |-  ( d  =  c  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  c ) )
12 fveq2 5632 . . . . . 6  |-  ( d  =  c  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  c ) )
1312iuneq2d 4032 . . . . 5  |-  ( d  =  c  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  c )
)
1411, 13eqeq12d 2380 . . . 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 4560 . . . . . 6  |-  ( d  =  suc  c  ->  suc  d  =  suc  suc  c )
1615fveq2d 5636 . . . . 5  |-  ( d  =  suc  c  -> 
( ( U `  b ) `  suc  d )  =  ( ( U `  b
) `  suc  suc  c
) )
17 fveq2 5632 . . . . . 6  |-  ( d  =  suc  c  -> 
( ( U `  a ) `  d
)  =  ( ( U `  a ) `
 suc  c )
)
1817iuneq2d 4032 . . . . 5  |-  ( d  =  suc  c  ->  U_ a  e.  b 
( ( U `  a ) `  d
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) )
1916, 18eqeq12d 2380 . . . 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 4560 . . . . . 6  |-  ( d  =  B  ->  suc  d  =  suc  B )
2120fveq2d 5636 . . . . 5  |-  ( d  =  B  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  B ) )
22 fveq2 5632 . . . . . 6  |-  ( d  =  B  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  B ) )
2322iuneq2d 4032 . . . . 5  |-  ( d  =  B  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  B )
)
2421, 23eqeq12d 2380 . . . 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 4057 . . . . 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 8192 . . . . . 6  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. ( ( U `
 b ) `  (/) )
28 vex 2876 . . . . . . . 8  |-  b  e. 
_V
2926ituni0 8191 . . . . . . . 8  |-  ( b  e.  _V  ->  (
( U `  b
) `  (/) )  =  b )
3028, 29ax-mp 8 . . . . . . 7  |-  ( ( U `  b ) `
 (/) )  =  b
3130unieqi 3939 . . . . . 6  |-  U. (
( U `  b
) `  (/) )  = 
U. b
3227, 31eqtri 2386 . . . . 5  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. b
3326ituni0 8191 . . . . . 6  |-  ( a  e.  b  ->  (
( U `  a
) `  (/) )  =  a )
3433iuneq2i 4025 . . . . 5  |-  U_ a  e.  b  ( ( U `  a ) `  (/) )  =  U_ a  e.  b  a
3525, 32, 343eqtr4i 2396 . . . 4  |-  ( ( U `  b ) `
 suc  (/) )  = 
U_ a  e.  b  ( ( U `  a ) `  (/) )
3626itunisuc 8192 . . . . . 6  |-  ( ( U `  b ) `
 suc  suc  c )  =  U. ( ( U `  b ) `
 suc  c )
37 unieq 3938 . . . . . . 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 8192 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
3938a1i 10 . . . . . . . . 9  |-  ( a  e.  b  ->  (
( U `  a
) `  suc  c )  =  U. ( ( U `  a ) `
 c ) )
4039iuneq2i 4025 . . . . . . . 8  |-  U_ a  e.  b  ( ( U `  a ) `  suc  c )  = 
U_ a  e.  b 
U. ( ( U `
 a ) `  c )
41 iuncom4 4014 . . . . . . . 8  |-  U_ a  e.  b  U. (
( U `  a
) `  c )  =  U. U_ a  e.  b  ( ( U `
 a ) `  c )
4240, 41eqtr2i 2387 . . . . . . 7  |-  U. U_ a  e.  b  (
( U `  a
) `  c )  =  U_ a  e.  b  ( ( U `  a ) `  suc  c )
4337, 42syl6eq 2414 . . . . . 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 2410 . . . . 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 10 . . . 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 4785 . . 3  |-  ( B  e.  om  ->  (
( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
47 iun0 4060 . . . . 5  |-  U_ a  e.  b  (/)  =  (/)
4847eqcomi 2370 . . . 4  |-  (/)  =  U_ a  e.  b  (/)
49 peano2b 4775 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
5026itunifn 8190 . . . . . . . 8  |-  ( b  e.  _V  ->  ( U `  b )  Fn  om )
51 fndm 5448 . . . . . . . 8  |-  ( ( U `  b )  Fn  om  ->  dom  ( U `  b )  =  om )
5228, 50, 51mp2b 9 . . . . . . 7  |-  dom  ( U `  b )  =  om
5352eleq2i 2430 . . . . . 6  |-  ( suc 
B  e.  dom  ( U `  b )  <->  suc 
B  e.  om )
5449, 53bitr4i 243 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
dom  ( U `  b ) )
55 ndmfv 5659 . . . . 5  |-  ( -. 
suc  B  e.  dom  ( U `  b )  ->  ( ( U `
 b ) `  suc  B )  =  (/) )
5654, 55sylnbi 297 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  (/) )
57 vex 2876 . . . . . . . 8  |-  a  e. 
_V
5826itunifn 8190 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
59 fndm 5448 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
6057, 58, 59mp2b 9 . . . . . . 7  |-  dom  ( U `  a )  =  om
6160eleq2i 2430 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
62 ndmfv 5659 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
6361, 62sylnbir 298 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
6463iuneq2d 4032 . . . 4  |-  ( -.  B  e.  om  ->  U_ a  e.  b  ( ( U `  a
) `  B )  =  U_ a  e.  b  (/) )
6548, 56, 643eqtr4a 2424 . . 3  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
6646, 65pm2.61i 156 . 2  |-  ( ( U `  b ) `
 suc  B )  =  U_ a  e.  b  ( ( U `  a ) `  B
)
674, 66vtoclg 2928 1  |-  ( A  e.  V  ->  (
( U `  A
) `  suc  B )  =  U_ a  e.  A  ( ( U `
 a ) `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873   (/)c0 3543   U.cuni 3929   U_ciun 4007    e. cmpt 4179   suc csuc 4497   omcom 4759   dom cdm 4792    |` cres 4794    Fn wfn 5353   ` cfv 5358   reccrdg 6564
This theorem is referenced by:  hsmexlem4  8202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-recs 6530  df-rdg 6565
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