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Theorem disjrnmpt2 37534
Description: Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
disjrnmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
disjrnmpt2  |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  F  y )
Distinct variable groups:    x, A    y, F
Allowed substitution hints:    A( y)    B( x, y)    F( x)

Proof of Theorem disjrnmpt2
Dummy variables  u  z  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7  |-  ( y  =  w  ->  y  =  w )
21cbvdisjv 4377 . . . . . 6  |-  (Disj  y  e.  ran  F  y  <-> Disj  w  e.  ran  F  w )
32notbii 303 . . . . 5  |-  ( -. Disj  y  e.  ran  F  y  <->  -. Disj  w  e.  ran  F  w )
4 id 22 . . . . . . 7  |-  ( w  =  v  ->  w  =  v )
54ndisj2 37448 . . . . . 6  |-  ( -. Disj  w  e.  ran  F  w  <->  E. w  e.  ran  F E. v  e.  ran  F ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
65biimpi 199 . . . . 5  |-  ( -. Disj  w  e.  ran  F  w  ->  E. w  e.  ran  F E. v  e.  ran  F ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
73, 6sylbi 200 . . . 4  |-  ( -. Disj  y  e.  ran  F  y  ->  E. w  e.  ran  F E. v  e.  ran  F ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
8 disjrnmpt2.1 . . . . . . . . . . . . . 14  |-  F  =  ( x  e.  A  |->  B )
98elrnmpt 5087 . . . . . . . . . . . . 13  |-  ( w  e.  ran  F  -> 
( w  e.  ran  F  <->  E. x  e.  A  w  =  B )
)
109ibi 249 . . . . . . . . . . . 12  |-  ( w  e.  ran  F  ->  E. x  e.  A  w  =  B )
1110adantr 472 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. x  e.  A  w  =  B )
12 nfcv 2612 . . . . . . . . . . . . . . . 16  |-  F/_ z B
13 nfcsb1v 3365 . . . . . . . . . . . . . . . 16  |-  F/_ x [_ z  /  x ]_ B
14 csbeq1a 3358 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1512, 13, 14cbvmpt 4487 . . . . . . . . . . . . . . 15  |-  ( x  e.  A  |->  B )  =  ( z  e.  A  |->  [_ z  /  x ]_ B )
168, 15eqtri 2493 . . . . . . . . . . . . . 14  |-  F  =  ( z  e.  A  |-> 
[_ z  /  x ]_ B )
1716elrnmpt 5087 . . . . . . . . . . . . 13  |-  ( v  e.  ran  F  -> 
( v  e.  ran  F  <->  E. z  e.  A  v  =  [_ z  /  x ]_ B ) )
1817ibi 249 . . . . . . . . . . . 12  |-  ( v  e.  ran  F  ->  E. z  e.  A  v  =  [_ z  /  x ]_ B )
1918adantl 473 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. z  e.  A  v  =  [_ z  /  x ]_ B )
2011, 19jca 541 . . . . . . . . . 10  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  ( E. x  e.  A  w  =  B  /\  E. z  e.  A  v  =  [_ z  /  x ]_ B ) )
21 nfv 1769 . . . . . . . . . . 11  |-  F/ z  w  =  B
22 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ x
v
2322, 13nfeq 2623 . . . . . . . . . . 11  |-  F/ x  v  =  [_ z  /  x ]_ B
2421, 23reean 2943 . . . . . . . . . 10  |-  ( E. x  e.  A  E. z  e.  A  (
w  =  B  /\  v  =  [_ z  /  x ]_ B )  <->  ( E. x  e.  A  w  =  B  /\  E. z  e.  A  v  =  [_ z  /  x ]_ B ) )
2520, 24sylibr 217 . . . . . . . . 9  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. x  e.  A  E. z  e.  A  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )
2625adantr 472 . . . . . . . 8  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  ->  E. x  e.  A  E. z  e.  A  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )
27 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ x w
28 nfmpt1 4485 . . . . . . . . . . . . . 14  |-  F/_ x
( x  e.  A  |->  B )
298, 28nfcxfr 2610 . . . . . . . . . . . . 13  |-  F/_ x F
3029nfrn 5083 . . . . . . . . . . . 12  |-  F/_ x ran  F
3127, 30nfel 2624 . . . . . . . . . . 11  |-  F/ x  w  e.  ran  F
3230nfcri 2606 . . . . . . . . . . 11  |-  F/ x  v  e.  ran  F
3331, 32nfan 2031 . . . . . . . . . 10  |-  F/ x
( w  e.  ran  F  /\  v  e.  ran  F )
34 nfv 1769 . . . . . . . . . 10  |-  F/ x
( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )
3533, 34nfan 2031 . . . . . . . . 9  |-  F/ x
( ( w  e. 
ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
36 simpll 768 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  w  =  B )
3714adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  B  =  [_ z  /  x ]_ B )
38 id 22 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  [_ z  /  x ]_ B  ->  v  =  [_ z  /  x ]_ B )
3938eqcomd 2477 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  [_ z  /  x ]_ B  ->  [_ z  /  x ]_ B  =  v )
4039ad2antlr 741 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  [_ z  /  x ]_ B  =  v )
4136, 37, 403eqtrd 2509 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  w  =  v )
4241adantll 728 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  w  =  v )
43 simpll 768 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  w  =/=  v )
4443neneqd 2648 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  -.  w  =  v
)
4542, 44pm2.65da 586 . . . . . . . . . . . . . . . 16  |-  ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  -.  x  =  z )
4645neqned 2650 . . . . . . . . . . . . . . 15  |-  ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  x  =/=  z )
4746adantlr 729 . . . . . . . . . . . . . 14  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  x  =/=  z )
48 id 22 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  B  ->  w  =  B )
4948eqcomd 2477 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  B  ->  B  =  w )
5049ad2antrl 742 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  B  =  w )
5139ad2antll 743 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  [_ z  /  x ]_ B  =  v
)
5250, 51ineq12d 3626 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =  ( w  i^i  v ) )
53 simpl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( w  i^i  v )  =/=  (/) )
5452, 53eqnetrd 2710 . . . . . . . . . . . . . . 15  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
5554adantll 728 . . . . . . . . . . . . . 14  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
5647, 55jca 541 . . . . . . . . . . . . 13  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
5756ex 441 . . . . . . . . . . . 12  |-  ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  -> 
( ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  (
x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
5857adantl 473 . . . . . . . . . . 11  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  -> 
( ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  (
x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
5958reximdv 2857 . . . . . . . . . 10  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  -> 
( E. z  e.  A  ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
6059a1d 25 . . . . . . . . 9  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  -> 
( x  e.  A  ->  ( E. z  e.  A  ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) ) )
6135, 60reximdai 2853 . . . . . . . 8  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  -> 
( E. x  e.  A  E. z  e.  A  ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
6226, 61mpd 15 . . . . . . 7  |-  ( ( ( w  e.  ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) ) )  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
6362ex 441 . . . . . 6  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  ( (
w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
6463a1i 11 . . . . 5  |-  ( -. Disj  y  e.  ran  F  y  ->  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  ( ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) )  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) ) )
6564rexlimdvv 2877 . . . 4  |-  ( -. Disj  y  e.  ran  F  y  ->  ( E. w  e.  ran  F E. v  e.  ran  F ( w  =/=  v  /\  (
w  i^i  v )  =/=  (/) )  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
667, 65mpd 15 . . 3  |-  ( -. Disj  y  e.  ran  F  y  ->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
67 nfcv 2612 . . . . . 6  |-  F/_ u B
68 nfcsb1v 3365 . . . . . 6  |-  F/_ x [_ u  /  x ]_ B
69 csbeq1a 3358 . . . . . 6  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
7067, 68, 69cbvdisj 4376 . . . . 5  |-  (Disj  x  e.  A  B  <-> Disj  u  e.  A  [_ u  /  x ]_ B )
7170notbii 303 . . . 4  |-  ( -. Disj  x  e.  A  B  <->  -. Disj  u  e.  A  [_ u  /  x ]_ B )
72 csbeq1a 3358 . . . . . . 7  |-  ( u  =  z  ->  [_ u  /  x ]_ B  = 
[_ z  /  u ]_ [_ u  /  x ]_ B )
73 csbco 3359 . . . . . . . 8  |-  [_ z  /  u ]_ [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B
7473a1i 11 . . . . . . 7  |-  ( u  =  z  ->  [_ z  /  u ]_ [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B )
7572, 74eqtrd 2505 . . . . . 6  |-  ( u  =  z  ->  [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B )
7675ndisj2 37448 . . . . 5  |-  ( -. Disj  u  e.  A  [_ u  /  x ]_ B  <->  E. u  e.  A  E. z  e.  A  ( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
77 nfcv 2612 . . . . . . 7  |-  F/_ x A
78 nfv 1769 . . . . . . . 8  |-  F/ x  u  =/=  z
7968, 13nfin 3630 . . . . . . . . 9  |-  F/_ x
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )
80 nfcv 2612 . . . . . . . . 9  |-  F/_ x (/)
8179, 80nfne 2742 . . . . . . . 8  |-  F/ x
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/)
8278, 81nfan 2031 . . . . . . 7  |-  F/ x
( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
8377, 82nfrex 2848 . . . . . 6  |-  F/ x E. z  e.  A  ( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
84 nfv 1769 . . . . . 6  |-  F/ u E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
85 neeq1 2705 . . . . . . . 8  |-  ( u  =  x  ->  (
u  =/=  z  <->  x  =/=  z ) )
86 csbeq1 3352 . . . . . . . . . . 11  |-  ( u  =  x  ->  [_ u  /  x ]_ B  = 
[_ x  /  x ]_ B )
87 csbid 3357 . . . . . . . . . . . 12  |-  [_ x  /  x ]_ B  =  B
8887a1i 11 . . . . . . . . . . 11  |-  ( u  =  x  ->  [_ x  /  x ]_ B  =  B )
8986, 88eqtrd 2505 . . . . . . . . . 10  |-  ( u  =  x  ->  [_ u  /  x ]_ B  =  B )
9089ineq1d 3624 . . . . . . . . 9  |-  ( u  =  x  ->  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  [_ z  /  x ]_ B ) )
9190neeq1d 2702 . . . . . . . 8  |-  ( u  =  x  ->  (
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/)  <->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9285, 91anbi12d 725 . . . . . . 7  |-  ( u  =  x  ->  (
( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )  <->  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
9392rexbidv 2892 . . . . . 6  |-  ( u  =  x  ->  ( E. z  e.  A  ( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )  <->  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
9483, 84, 93cbvrex 3002 . . . . 5  |-  ( E. u  e.  A  E. z  e.  A  (
u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )  <->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9576, 94bitri 257 . . . 4  |-  ( -. Disj  u  e.  A  [_ u  /  x ]_ B  <->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9671, 95bitri 257 . . 3  |-  ( -. Disj  x  e.  A  B  <->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9766, 96sylibr 217 . 2  |-  ( -. Disj  y  e.  ran  F  y  ->  -. Disj  x  e.  A  B )
9897con4i 135 1  |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  F  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   [_csb 3349    i^i cin 3389   (/)c0 3722  Disj wdisj 4366    |-> cmpt 4454   ran crn 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-cnv 4847  df-dm 4849  df-rn 4850
This theorem is referenced by:  meadjiun  38420
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