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Theorem disjabrexf 23978
Description: Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
disjabrexf.1  |-  F/_ x A
Assertion
Ref Expression
disjabrexf  |-  (Disj  x  e.  A B  -> Disj  y  e. 
{ z  |  E. x  e.  A  z  =  B } y )
Distinct variable groups:    x, y,
z    y, A, z    y, B, z
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem disjabrexf
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 4155 . . . 4  |-  F/ xDisj  x  e.  A B
2 nfcv 2540 . . . . 5  |-  F/_ x
y
3 disjabrexf.1 . . . . . . . . . . 11  |-  F/_ x A
43nfcri 2534 . . . . . . . . . 10  |-  F/ x  i  e.  A
5 nfcsb1v 3243 . . . . . . . . . . 11  |-  F/_ x [_ i  /  x ]_ B
65nfcri 2534 . . . . . . . . . 10  |-  F/ x  j  e.  [_ i  /  x ]_ B
74, 6nfan 1842 . . . . . . . . 9  |-  F/ x
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )
87nfab 2544 . . . . . . . 8  |-  F/_ x { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }
98nfuni 3981 . . . . . . 7  |-  F/_ x U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }
109nfcsb1 3242 . . . . . 6  |-  F/_ x [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B
1110nfeq1 2549 . . . . 5  |-  F/ x [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y
122, 11nfral 2719 . . . 4  |-  F/ x A. j  e.  y  [_ U. { i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  /  x ]_ B  =  y
13 eqeq2 2413 . . . . 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 ) )
1413raleqbi1dv 2872 . . . 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 ) )
15 vex 2919 . . . . 5  |-  y  e. 
_V
1615a1i 11 . . . 4  |-  (Disj  x  e.  A B  ->  y  e.  _V )
17 simplll 735 . . . . . . . . . . . . 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 )
18 simpllr 736 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  e.  A
)
19 simprl 733 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  i  e.  A
)
20 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  j  e.  B
)
21 simprr 734 . . . . . . . . . . . . 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
)
22 csbeq1a 3219 . . . . . . . . . . . . . 14  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
233, 5, 22disjif2 23976 . . . . . . . . . . . . 13  |-  ( (Disj  x  e.  A B  /\  ( x  e.  A  /\  i  e.  A
)  /\  ( j  e.  B  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  =  i )
2417, 18, 19, 20, 21, 23syl122anc 1193 . . . . . . . . . . . 12  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  (
i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )  ->  x  =  i )
25 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  ->  x  =  i )
26 simpllr 736 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  ->  x  e.  A )
2725, 26eqeltrrd 2479 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
i  e.  A )
28 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
j  e.  B )
2922eleq2d 2471 . . . . . . . . . . . . . . 15  |-  ( x  =  i  ->  (
j  e.  B  <->  j  e.  [_ i  /  x ]_ B ) )
3025, 29syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
( j  e.  B  <->  j  e.  [_ i  /  x ]_ B ) )
3128, 30mpbid 202 . . . . . . . . . . . . 13  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
j  e.  [_ i  /  x ]_ B )
3227, 31jca 519 . . . . . . . . . . . 12  |-  ( ( ( (Disj  x  e.  A B  /\  x  e.  A )  /\  j  e.  B )  /\  x  =  i )  -> 
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) )
3324, 32impbida 806 . . . . . . . . . . 11  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  (
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )  <-> 
x  =  i ) )
34 equcom 1688 . . . . . . . . . . 11  |-  ( x  =  i  <->  i  =  x )
3533, 34syl6bb 253 . . . . . . . . . 10  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  (
( i  e.  A  /\  j  e.  [_ i  /  x ]_ B )  <-> 
i  =  x ) )
3635abbidv 2518 . . . . . . . . 9  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  { i  |  ( i  e.  A  /\  j  e. 
[_ i  /  x ]_ B ) }  =  { i  |  i  =  x } )
37 df-sn 3780 . . . . . . . . 9  |-  { x }  =  { i  |  i  =  x }
3836, 37syl6eqr 2454 . . . . . . . 8  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  { i  |  ( i  e.  A  /\  j  e. 
[_ i  /  x ]_ B ) }  =  { x } )
3938unieqd 3986 . . . . . . 7  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  U. { x } )
40 vex 2919 . . . . . . . 8  |-  x  e. 
_V
4140unisn 3991 . . . . . . 7  |-  U. {
x }  =  x
4239, 41syl6eq 2452 . . . . . 6  |-  ( ( (Disj  x  e.  A B  /\  x  e.  A
)  /\  j  e.  B )  ->  U. {
i  |  ( i  e.  A  /\  j  e.  [_ i  /  x ]_ B ) }  =  x )
43 csbeq1 3214 . . . . . . 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 )
44 csbid 3218 . . . . . . 7  |-  [_ x  /  x ]_ B  =  B
4543, 44syl6eq 2452 . . . . . 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 )
4642, 45syl 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 )
4746ralrimiva 2749 . . . 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 )
481, 12, 14, 16, 47elabreximd 23944 . . 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 )
4948ralrimiva 2749 . 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 )
50 invdisj 4161 . 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 )
5149, 50syl 16 1  |-  (Disj  x  e.  A B  -> Disj  y  e. 
{ z  |  E. x  e.  A  z  =  B } y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   F/_wnfc 2527   A.wral 2666   E.wrex 2667   _Vcvv 2916   [_csb 3211   {csn 3774   U.cuni 3975  Disj wdisj 4142
This theorem is referenced by:  measvunilem  24519
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rmo 2674  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-nul 3589  df-sn 3780  df-pr 3781  df-uni 3976  df-disj 4143
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