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Theorem disjrnmpt2 37313
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 23 . . . . . . 7  |-  ( y  =  w  ->  y  =  w )
21cbvdisjv 4402 . . . . . 6  |-  (Disj  y  e.  ran  F  y  <-> Disj  w  e.  ran  F  w )
32notbii 297 . . . . 5  |-  ( -. Disj  y  e.  ran  F  y  <->  -. Disj  w  e.  ran  F  w )
4 id 23 . . . . . . 7  |-  ( w  =  v  ->  w  =  v )
54ndisj2 37251 . . . . . 6  |-  ( -. Disj  w  e.  ran  F  w  <->  E. w  e.  ran  F E. v  e.  ran  F ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
65biimpi 197 . . . . 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 198 . . . 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 5097 . . . . . . . . . . . . 13  |-  ( w  e.  ran  F  -> 
( w  e.  ran  F  <->  E. x  e.  A  w  =  B )
)
109ibi 244 . . . . . . . . . . . 12  |-  ( w  e.  ran  F  ->  E. x  e.  A  w  =  B )
1110adantr 466 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. x  e.  A  w  =  B )
12 nfcv 2584 . . . . . . . . . . . . . . . 16  |-  F/_ z B
13 nfcsb1v 3411 . . . . . . . . . . . . . . . 16  |-  F/_ x [_ z  /  x ]_ B
14 csbeq1a 3404 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1512, 13, 14cbvmpt 4512 . . . . . . . . . . . . . . 15  |-  ( x  e.  A  |->  B )  =  ( z  e.  A  |->  [_ z  /  x ]_ B )
168, 15eqtri 2451 . . . . . . . . . . . . . 14  |-  F  =  ( z  e.  A  |-> 
[_ z  /  x ]_ B )
1716elrnmpt 5097 . . . . . . . . . . . . 13  |-  ( v  e.  ran  F  -> 
( v  e.  ran  F  <->  E. z  e.  A  v  =  [_ z  /  x ]_ B ) )
1817ibi 244 . . . . . . . . . . . 12  |-  ( v  e.  ran  F  ->  E. z  e.  A  v  =  [_ z  /  x ]_ B )
1918adantl 467 . . . . . . . . . . 11  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. z  e.  A  v  =  [_ z  /  x ]_ B )
2011, 19jca 534 . . . . . . . . . 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 1751 . . . . . . . . . . 11  |-  F/ z  w  =  B
22 nfcv 2584 . . . . . . . . . . . 12  |-  F/_ x
v
2322, 13nfeq 2595 . . . . . . . . . . 11  |-  F/ x  v  =  [_ z  /  x ]_ B
2421, 23reean 2995 . . . . . . . . . 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 215 . . . . . . . . 9  |-  ( ( w  e.  ran  F  /\  v  e.  ran  F )  ->  E. x  e.  A  E. z  e.  A  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )
2625adantr 466 . . . . . . . 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 2584 . . . . . . . . . . . 12  |-  F/_ x w
28 nfmpt1 4510 . . . . . . . . . . . . . 14  |-  F/_ x
( x  e.  A  |->  B )
298, 28nfcxfr 2582 . . . . . . . . . . . . 13  |-  F/_ x F
3029nfrn 5093 . . . . . . . . . . . 12  |-  F/_ x ran  F
3127, 30nfel 2597 . . . . . . . . . . 11  |-  F/ x  w  e.  ran  F
3230nfcri 2577 . . . . . . . . . . 11  |-  F/ x  v  e.  ran  F
3331, 32nfan 1984 . . . . . . . . . 10  |-  F/ x
( w  e.  ran  F  /\  v  e.  ran  F )
34 nfv 1751 . . . . . . . . . 10  |-  F/ x
( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )
3533, 34nfan 1984 . . . . . . . . 9  |-  F/ x
( ( w  e. 
ran  F  /\  v  e.  ran  F )  /\  ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) ) )
36 simpll 758 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  w  =  B )
3714adantl 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  B  =  [_ z  /  x ]_ B )
38 id 23 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  =  [_ z  /  x ]_ B  ->  v  =  [_ z  /  x ]_ B )
3938eqcomd 2430 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  [_ z  /  x ]_ B  ->  [_ z  /  x ]_ B  =  v )
4039ad2antlr 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  [_ z  /  x ]_ B  =  v )
4136, 37, 403eqtrd 2467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  =  B  /\  v  =  [_ z  /  x ]_ B
)  /\  x  =  z )  ->  w  =  v )
4241adantll 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  w  =  v )
43 simpll 758 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  w  =/=  v )
4443neneqd 2625 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B
) )  /\  x  =  z )  ->  -.  w  =  v
)
4542, 44pm2.65da 578 . . . . . . . . . . . . . . . 16  |-  ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  -.  x  =  z )
4645neqned 2627 . . . . . . . . . . . . . . 15  |-  ( ( w  =/=  v  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  x  =/=  z )
4746adantlr 719 . . . . . . . . . . . . . 14  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  x  =/=  z )
48 id 23 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  B  ->  w  =  B )
4948eqcomd 2430 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  B  ->  B  =  w )
5049ad2antrl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  B  =  w )
5139ad2antll 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  [_ z  /  x ]_ B  =  v
)
5250, 51ineq12d 3665 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =  ( w  i^i  v ) )
53 simpl 458 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( w  i^i  v )  =/=  (/) )
5452, 53eqnetrd 2717 . . . . . . . . . . . . . . 15  |-  ( ( ( w  i^i  v
)  =/=  (/)  /\  (
w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
5554adantll 718 . . . . . . . . . . . . . 14  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
5647, 55jca 534 . . . . . . . . . . . . 13  |-  ( ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  /\  ( w  =  B  /\  v  =  [_ z  /  x ]_ B ) )  ->  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
5756ex 435 . . . . . . . . . . . 12  |-  ( ( w  =/=  v  /\  ( w  i^i  v
)  =/=  (/) )  -> 
( ( w  =  B  /\  v  = 
[_ z  /  x ]_ B )  ->  (
x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
5857adantl 467 . . . . . . . . . . 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 2899 . . . . . . . . . 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 26 . . . . . . . . 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 2894 . . . . . . . 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 435 . . . . . 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 2923 . . . 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 2584 . . . . . 6  |-  F/_ u B
68 nfcsb1v 3411 . . . . . 6  |-  F/_ x [_ u  /  x ]_ B
69 csbeq1a 3404 . . . . . 6  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
7067, 68, 69cbvdisj 4401 . . . . 5  |-  (Disj  x  e.  A  B  <-> Disj  u  e.  A  [_ u  /  x ]_ B )
7170notbii 297 . . . 4  |-  ( -. Disj  x  e.  A  B  <->  -. Disj  u  e.  A  [_ u  /  x ]_ B )
72 csbeq1a 3404 . . . . . . 7  |-  ( u  =  z  ->  [_ u  /  x ]_ B  = 
[_ z  /  u ]_ [_ u  /  x ]_ B )
73 csbco 3405 . . . . . . . 8  |-  [_ z  /  u ]_ [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B
7473a1i 11 . . . . . . 7  |-  ( u  =  z  ->  [_ z  /  u ]_ [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B )
7572, 74eqtrd 2463 . . . . . 6  |-  ( u  =  z  ->  [_ u  /  x ]_ B  = 
[_ z  /  x ]_ B )
7675ndisj2 37251 . . . . 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 2584 . . . . . . 7  |-  F/_ x A
78 nfv 1751 . . . . . . . 8  |-  F/ x  u  =/=  z
7968, 13nfin 3669 . . . . . . . . 9  |-  F/_ x
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )
80 nfcv 2584 . . . . . . . . 9  |-  F/_ x (/)
8179, 80nfne 2756 . . . . . . . 8  |-  F/ x
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/)
8278, 81nfan 1984 . . . . . . 7  |-  F/ x
( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
8377, 82nfrex 2888 . . . . . 6  |-  F/ x E. z  e.  A  ( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )
84 nfv 1751 . . . . . 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 3398 . . . . . . . . . . 11  |-  ( u  =  x  ->  [_ u  /  x ]_ B  = 
[_ x  /  x ]_ B )
87 csbid 3403 . . . . . . . . . . . 12  |-  [_ x  /  x ]_ B  =  B
8887a1i 11 . . . . . . . . . . 11  |-  ( u  =  x  ->  [_ x  /  x ]_ B  =  B )
8986, 88eqtrd 2463 . . . . . . . . . 10  |-  ( u  =  x  ->  [_ u  /  x ]_ B  =  B )
9089ineq1d 3663 . . . . . . . . 9  |-  ( u  =  x  ->  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =  ( B  i^i  [_ z  /  x ]_ B ) )
9190neeq1d 2701 . . . . . . . 8  |-  ( u  =  x  ->  (
( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/)  <->  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9285, 91anbi12d 715 . . . . . . 7  |-  ( u  =  x  ->  (
( u  =/=  z  /\  ( [_ u  /  x ]_ B  i^i  [_ z  /  x ]_ B )  =/=  (/) )  <->  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) ) )
9392rexbidv 2939 . . . . . 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 3052 . . . . 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 252 . . . 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 252 . . 3  |-  ( -. Disj  x  e.  A  B  <->  E. x  e.  A  E. z  e.  A  ( x  =/=  z  /\  ( B  i^i  [_ z  /  x ]_ B )  =/=  (/) ) )
9766, 96sylibr 215 . 2  |-  ( -. Disj  y  e.  ran  F  y  ->  -. Disj  x  e.  A  B )
9897con4i 133 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 370    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   [_csb 3395    i^i cin 3435   (/)c0 3761  Disj wdisj 4391    |-> cmpt 4479   ran crn 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-cnv 4858  df-dm 4860  df-rn 4861
This theorem is referenced by:  meadjiun  38083
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