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Theorem ituniiun 8266
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 5695 . . . 4  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
21fveq1d 5697 . . 3  |-  ( b  =  A  ->  (
( U `  b
) `  suc  B )  =  ( ( U `
 A ) `  suc  B ) )
3 iuneq1 4074 . . 3  |-  ( b  =  A  ->  U_ a  e.  b  ( ( U `  a ) `  B )  =  U_ a  e.  A  (
( U `  a
) `  B )
)
42, 3eqeq12d 2426 . 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 4614 . . . . . 6  |-  ( d  =  (/)  ->  suc  d  =  suc  (/) )
65fveq2d 5699 . . . . 5  |-  ( d  =  (/)  ->  ( ( U `  b ) `
 suc  d )  =  ( ( U `
 b ) `  suc  (/) ) )
7 fveq2 5695 . . . . . 6  |-  ( d  =  (/)  ->  ( ( U `  a ) `
 d )  =  ( ( U `  a ) `  (/) ) )
87iuneq2d 4086 . . . . 5  |-  ( d  =  (/)  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) )
96, 8eqeq12d 2426 . . . 4  |-  ( d  =  (/)  ->  ( ( ( U `  b
) `  suc  d )  =  U_ a  e.  b  ( ( U `
 a ) `  d )  <->  ( ( U `  b ) `  suc  (/) )  =  U_ a  e.  b  (
( U `  a
) `  (/) ) ) )
10 suceq 4614 . . . . . 6  |-  ( d  =  c  ->  suc  d  =  suc  c )
1110fveq2d 5699 . . . . 5  |-  ( d  =  c  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  c ) )
12 fveq2 5695 . . . . . 6  |-  ( d  =  c  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  c ) )
1312iuneq2d 4086 . . . . 5  |-  ( d  =  c  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  c )
)
1411, 13eqeq12d 2426 . . . 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 4614 . . . . . 6  |-  ( d  =  suc  c  ->  suc  d  =  suc  suc  c )
1615fveq2d 5699 . . . . 5  |-  ( d  =  suc  c  -> 
( ( U `  b ) `  suc  d )  =  ( ( U `  b
) `  suc  suc  c
) )
17 fveq2 5695 . . . . . 6  |-  ( d  =  suc  c  -> 
( ( U `  a ) `  d
)  =  ( ( U `  a ) `
 suc  c )
)
1817iuneq2d 4086 . . . . 5  |-  ( d  =  suc  c  ->  U_ a  e.  b 
( ( U `  a ) `  d
)  =  U_ a  e.  b  ( ( U `  a ) `  suc  c ) )
1916, 18eqeq12d 2426 . . . 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 4614 . . . . . 6  |-  ( d  =  B  ->  suc  d  =  suc  B )
2120fveq2d 5699 . . . . 5  |-  ( d  =  B  ->  (
( U `  b
) `  suc  d )  =  ( ( U `
 b ) `  suc  B ) )
22 fveq2 5695 . . . . . 6  |-  ( d  =  B  ->  (
( U `  a
) `  d )  =  ( ( U `
 a ) `  B ) )
2322iuneq2d 4086 . . . . 5  |-  ( d  =  B  ->  U_ a  e.  b  ( ( U `  a ) `  d )  =  U_ a  e.  b  (
( U `  a
) `  B )
)
2421, 23eqeq12d 2426 . . . 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 4112 . . . . 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 8263 . . . . . 6  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. ( ( U `
 b ) `  (/) )
28 vex 2927 . . . . . . . 8  |-  b  e. 
_V
2926ituni0 8262 . . . . . . . 8  |-  ( b  e.  _V  ->  (
( U `  b
) `  (/) )  =  b )
3028, 29ax-mp 8 . . . . . . 7  |-  ( ( U `  b ) `
 (/) )  =  b
3130unieqi 3993 . . . . . 6  |-  U. (
( U `  b
) `  (/) )  = 
U. b
3227, 31eqtri 2432 . . . . 5  |-  ( ( U `  b ) `
 suc  (/) )  = 
U. b
3326ituni0 8262 . . . . . 6  |-  ( a  e.  b  ->  (
( U `  a
) `  (/) )  =  a )
3433iuneq2i 4079 . . . . 5  |-  U_ a  e.  b  ( ( U `  a ) `  (/) )  =  U_ a  e.  b  a
3525, 32, 343eqtr4i 2442 . . . 4  |-  ( ( U `  b ) `
 suc  (/) )  = 
U_ a  e.  b  ( ( U `  a ) `  (/) )
3626itunisuc 8263 . . . . . 6  |-  ( ( U `  b ) `
 suc  suc  c )  =  U. ( ( U `  b ) `
 suc  c )
37 unieq 3992 . . . . . . 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 8263 . . . . . . . . . 10  |-  ( ( U `  a ) `
 suc  c )  =  U. ( ( U `
 a ) `  c )
3938a1i 11 . . . . . . . . 9  |-  ( a  e.  b  ->  (
( U `  a
) `  suc  c )  =  U. ( ( U `  a ) `
 c ) )
4039iuneq2i 4079 . . . . . . . 8  |-  U_ a  e.  b  ( ( U `  a ) `  suc  c )  = 
U_ a  e.  b 
U. ( ( U `
 a ) `  c )
41 iuncom4 4068 . . . . . . . 8  |-  U_ a  e.  b  U. (
( U `  a
) `  c )  =  U. U_ a  e.  b  ( ( U `
 a ) `  c )
4240, 41eqtr2i 2433 . . . . . . 7  |-  U. U_ a  e.  b  (
( U `  a
) `  c )  =  U_ a  e.  b  ( ( U `  a ) `  suc  c )
4337, 42syl6eq 2460 . . . . . 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 2456 . . . . 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 4838 . . 3  |-  ( B  e.  om  ->  (
( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
47 iun0 4115 . . . . 5  |-  U_ a  e.  b  (/)  =  (/)
4847eqcomi 2416 . . . 4  |-  (/)  =  U_ a  e.  b  (/)
49 peano2b 4828 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
5026itunifn 8261 . . . . . . . 8  |-  ( b  e.  _V  ->  ( U `  b )  Fn  om )
51 fndm 5511 . . . . . . . 8  |-  ( ( U `  b )  Fn  om  ->  dom  ( U `  b )  =  om )
5228, 50, 51mp2b 10 . . . . . . 7  |-  dom  ( U `  b )  =  om
5352eleq2i 2476 . . . . . 6  |-  ( suc 
B  e.  dom  ( U `  b )  <->  suc 
B  e.  om )
5449, 53bitr4i 244 . . . . 5  |-  ( B  e.  om  <->  suc  B  e. 
dom  ( U `  b ) )
55 ndmfv 5722 . . . . 5  |-  ( -. 
suc  B  e.  dom  ( U `  b )  ->  ( ( U `
 b ) `  suc  B )  =  (/) )
5654, 55sylnbi 298 . . . 4  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  (/) )
57 vex 2927 . . . . . . . 8  |-  a  e. 
_V
5826itunifn 8261 . . . . . . . 8  |-  ( a  e.  _V  ->  ( U `  a )  Fn  om )
59 fndm 5511 . . . . . . . 8  |-  ( ( U `  a )  Fn  om  ->  dom  ( U `  a )  =  om )
6057, 58, 59mp2b 10 . . . . . . 7  |-  dom  ( U `  a )  =  om
6160eleq2i 2476 . . . . . 6  |-  ( B  e.  dom  ( U `
 a )  <->  B  e.  om )
62 ndmfv 5722 . . . . . 6  |-  ( -.  B  e.  dom  ( U `  a )  ->  ( ( U `  a ) `  B
)  =  (/) )
6361, 62sylnbir 299 . . . . 5  |-  ( -.  B  e.  om  ->  ( ( U `  a
) `  B )  =  (/) )
6463iuneq2d 4086 . . . 4  |-  ( -.  B  e.  om  ->  U_ a  e.  b  ( ( U `  a
) `  B )  =  U_ a  e.  b  (/) )
6548, 56, 643eqtr4a 2470 . . 3  |-  ( -.  B  e.  om  ->  ( ( U `  b
) `  suc  B )  =  U_ a  e.  b  ( ( U `
 a ) `  B ) )
6646, 65pm2.61i 158 . 2  |-  ( ( U `  b ) `
 suc  B )  =  U_ a  e.  b  ( ( U `  a ) `  B
)
674, 66vtoclg 2979 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 1649    e. wcel 1721   _Vcvv 2924   (/)c0 3596   U.cuni 3983   U_ciun 4061    e. cmpt 4234   suc csuc 4551   omcom 4812   dom cdm 4845    |` cres 4847    Fn wfn 5416   ` cfv 5421   reccrdg 6634
This theorem is referenced by:  hsmexlem4  8273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635
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