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Theorem disjpreima 23979
Description: A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
disjpreima  |-  ( ( Fun  F  /\ Disj  x  e.  A B )  -> Disj  x  e.  A ( `' F " B ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    B( x)

Proof of Theorem disjpreima
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inpreima 5816 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' F " ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B ) )  =  ( ( `' F "
[_ y  /  x ]_ B )  i^i  ( `' F " [_ z  /  x ]_ B ) ) )
2 imaeq2 5158 . . . . . . . . . 10  |-  ( (
[_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  ->  ( `' F " ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B ) )  =  ( `' F " (/) ) )
3 ima0 5180 . . . . . . . . . 10  |-  ( `' F " (/) )  =  (/)
42, 3syl6eq 2452 . . . . . . . . 9  |-  ( (
[_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  ->  ( `' F " ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B ) )  =  (/) )
51, 4sylan9req 2457 . . . . . . . 8  |-  ( ( Fun  F  /\  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  ->  (
( `' F " [_ y  /  x ]_ B )  i^i  ( `' F " [_ z  /  x ]_ B ) )  =  (/) )
65ex 424 . . . . . . 7  |-  ( Fun 
F  ->  ( ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  ->  ( ( `' F " [_ y  /  x ]_ B )  i^i  ( `' F "
[_ z  /  x ]_ B ) )  =  (/) ) )
7 vex 2919 . . . . . . . . . . 11  |-  y  e. 
_V
8 csbima12g 5172 . . . . . . . . . . 11  |-  ( y  e.  _V  ->  [_ y  /  x ]_ ( `' F " B )  =  ( [_ y  /  x ]_ `' F "
[_ y  /  x ]_ B ) )
97, 8ax-mp 8 . . . . . . . . . 10  |-  [_ y  /  x ]_ ( `' F " B )  =  ( [_ y  /  x ]_ `' F "
[_ y  /  x ]_ B )
10 csbconstg 3225 . . . . . . . . . . . 12  |-  ( y  e.  _V  ->  [_ y  /  x ]_ `' F  =  `' F )
117, 10ax-mp 8 . . . . . . . . . . 11  |-  [_ y  /  x ]_ `' F  =  `' F
1211imaeq1i 5159 . . . . . . . . . 10  |-  ( [_ y  /  x ]_ `' F " [_ y  /  x ]_ B )  =  ( `' F " [_ y  /  x ]_ B )
139, 12eqtri 2424 . . . . . . . . 9  |-  [_ y  /  x ]_ ( `' F " B )  =  ( `' F "
[_ y  /  x ]_ B )
14 vex 2919 . . . . . . . . . . 11  |-  z  e. 
_V
15 csbima12g 5172 . . . . . . . . . . 11  |-  ( z  e.  _V  ->  [_ z  /  x ]_ ( `' F " B )  =  ( [_ z  /  x ]_ `' F "
[_ z  /  x ]_ B ) )
1614, 15ax-mp 8 . . . . . . . . . 10  |-  [_ z  /  x ]_ ( `' F " B )  =  ( [_ z  /  x ]_ `' F "
[_ z  /  x ]_ B )
17 csbconstg 3225 . . . . . . . . . . . 12  |-  ( z  e.  _V  ->  [_ z  /  x ]_ `' F  =  `' F )
1814, 17ax-mp 8 . . . . . . . . . . 11  |-  [_ z  /  x ]_ `' F  =  `' F
1918imaeq1i 5159 . . . . . . . . . 10  |-  ( [_ z  /  x ]_ `' F " [_ z  /  x ]_ B )  =  ( `' F " [_ z  /  x ]_ B )
2016, 19eqtri 2424 . . . . . . . . 9  |-  [_ z  /  x ]_ ( `' F " B )  =  ( `' F "
[_ z  /  x ]_ B )
2113, 20ineq12i 3500 . . . . . . . 8  |-  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  ( ( `' F "
[_ y  /  x ]_ B )  i^i  ( `' F " [_ z  /  x ]_ B ) )
2221eqeq1i 2411 . . . . . . 7  |-  ( (
[_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/)  <->  ( ( `' F " [_ y  /  x ]_ B )  i^i  ( `' F "
[_ z  /  x ]_ B ) )  =  (/) )
236, 22syl6ibr 219 . . . . . 6  |-  ( Fun 
F  ->  ( ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/)  ->  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/) ) )
2423orim2d 814 . . . . 5  |-  ( Fun 
F  ->  ( (
y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  ->  (
y  =  z  \/  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/) ) ) )
2524ralimdv 2745 . . . 4  |-  ( Fun 
F  ->  ( A. z  e.  A  (
y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  ->  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/) ) ) )
2625ralimdv 2745 . . 3  |-  ( Fun 
F  ->  ( A. y  e.  A  A. z  e.  A  (
y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) )  ->  A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/) ) ) )
27 disjors 4158 . . 3  |-  (Disj  x  e.  A B  <->  A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  (/) ) )
28 disjors 4158 . . 3  |-  (Disj  x  e.  A ( `' F " B )  <->  A. y  e.  A  A. z  e.  A  ( y  =  z  \/  ( [_ y  /  x ]_ ( `' F " B )  i^i  [_ z  /  x ]_ ( `' F " B ) )  =  (/) ) )
2926, 27, 283imtr4g 262 . 2  |-  ( Fun 
F  ->  (Disj  x  e.  A B  -> Disj  x  e.  A ( `' F " B ) ) )
3029imp 419 1  |-  ( ( Fun  F  /\ Disj  x  e.  A B )  -> Disj  x  e.  A ( `' F " B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [_csb 3211    i^i cin 3279   (/)c0 3588  Disj wdisj 4142   `'ccnv 4836   "cima 4840   Fun wfun 5407
This theorem is referenced by:  sibfof  24607  dstrvprob  24682
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-disj 4143  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415
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