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Theorem disjabrex 27582
Description: Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.)
Assertion
Ref Expression
disjabrex  |-  (Disj  x  e.  A  B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
Distinct variable groups:    x, y,
z, A    y, B, z
Allowed substitution hint:    B( x)

Proof of Theorem disjabrex
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 4440 . . . 4  |-  F/ xDisj  x  e.  A  B
2 nfcv 2619 . . . . 5  |-  F/_ x
y
3 nfv 1708 . . . . . . . . . 10  |-  F/ x  i  e.  A
4 nfcsb1v 3446 . . . . . . . . . . 11  |-  F/_ x [_ i  /  x ]_ B
54nfcri 2612 . . . . . . . . . 10  |-  F/ x  j  e.  [_ i  /  x ]_ B
63, 5nfan 1929 . . . . . . . . 9  |-  F/ x
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )
76nfab 2623 . . . . . . . 8  |-  F/_ x { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }
87nfuni 4257 . . . . . . 7  |-  F/_ x U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }
98nfcsb1 3445 . . . . . 6  |-  F/_ x [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B
109nfeq1 2634 . . . . 5  |-  F/ x [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y
112, 10nfral 2843 . . . 4  |-  F/ x A. j  e.  y  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y
12 eqeq2 2472 . . . . 5  |-  ( y  =  B  ->  ( [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y  <->  [_ U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  B ) )
1312raleqbi1dv 3062 . . . 4  |-  ( y  =  B  ->  ( A. j  e.  y  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y  <->  A. j  e.  B  [_ U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  B ) )
14 vex 3112 . . . . 5  |-  y  e. 
_V
1514a1i 11 . . . 4  |-  (Disj  x  e.  A  B  ->  y  e.  _V )
16 simplll 759 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  -> Disj  x  e.  A  B
)
17 simpllr 760 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  e.  A
)
18 simprl 756 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  i  e.  A
)
19 simplr 755 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  j  e.  B
)
20 simprr 757 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  j  e.  [_ i  /  x ]_ B
)
21 csbeq1a 3439 . . . . . . . . . . . . . 14  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
224, 21disjif 27578 . . . . . . . . . . . . 13  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  i  e.  A
)  /\  ( j  e.  B  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  =  i )
2316, 17, 18, 19, 20, 22syl122anc 1237 . . . . . . . . . . . 12  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  =  i )
24 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  ->  x  =  i )
25 simpllr 760 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  ->  x  e.  A )
2624, 25eqeltrrd 2546 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
i  e.  A )
27 simplr 755 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
j  e.  B )
2821eleq2d 2527 . . . . . . . . . . . . . . 15  |-  ( x  =  i  ->  (
j  e.  B  <->  j  e.  [_ i  /  x ]_ B ) )
2924, 28syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
( j  e.  B  <->  j  e.  [_ i  /  x ]_ B ) )
3027, 29mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
j  e.  [_ i  /  x ]_ B )
3126, 30jca 532 . . . . . . . . . . . 12  |-  ( ( ( (Disj  x  e.  A  B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )
3223, 31impbida 832 . . . . . . . . . . 11  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  (
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )  <-> 
x  =  i ) )
33 equcom 1795 . . . . . . . . . . 11  |-  ( x  =  i  <->  i  =  x )
3432, 33syl6bb 261 . . . . . . . . . 10  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  (
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )  <-> 
i  =  x ) )
3534abbidv 2593 . . . . . . . . 9  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  { i  |  ( i  e.  A  /\  j  e. 
[_ i  /  x ]_ B ) }  =  { i  |  i  =  x } )
36 df-sn 4033 . . . . . . . . 9  |-  { x }  =  { i  |  i  =  x }
3735, 36syl6eqr 2516 . . . . . . . 8  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  { i  |  ( i  e.  A  /\  j  e. 
[_ i  /  x ]_ B ) }  =  { x } )
3837unieqd 4261 . . . . . . 7  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  U. { x } )
39 vex 3112 . . . . . . . 8  |-  x  e. 
_V
4039unisn 4266 . . . . . . 7  |-  U. {
x }  =  x
4138, 40syl6eq 2514 . . . . . 6  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  x )
42 csbeq1 3433 . . . . . . 7  |-  ( U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  x  ->  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  [_ x  /  x ]_ B )
43 csbid 3438 . . . . . . 7  |-  [_ x  /  x ]_ B  =  B
4442, 43syl6eq 2514 . . . . . 6  |-  ( U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  x  ->  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  B )
4541, 44syl 16 . . . . 5  |-  ( ( (Disj  x  e.  A  B  /\  x  e.  A
)  /\  j  e.  B )  ->  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  B )
4645ralrimiva 2871 . . . 4  |-  ( (Disj  x  e.  A  B  /\  x  e.  A
)  ->  A. j  e.  B  [_ U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  B )
471, 11, 13, 15, 46elabreximd 27536 . . 3  |-  ( (Disj  x  e.  A  B  /\  y  e.  { z  |  E. x  e.  A  z  =  B } )  ->  A. j  e.  y  [_ U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y )
4847ralrimiva 2871 . 2  |-  (Disj  x  e.  A  B  ->  A. y  e.  { z  |  E. x  e.  A  z  =  B } A. j  e.  y  [_ U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y )
49 invdisj 4445 . 2  |-  ( A. y  e.  { z  |  E. x  e.  A  z  =  B } A. j  e.  y  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
5048, 49syl 16 1  |-  (Disj  x  e.  A  B  -> Disj  y  e.  { z  |  E. x  e.  A  z  =  B } y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109   [_csb 3430   {csn 4032   U.cuni 4251  Disj wdisj 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-nul 3794  df-sn 4033  df-pr 4035  df-uni 4252  df-disj 4428
This theorem is referenced by:  disjrnmpt  27585
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