MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnviin Unicode version

Theorem cnviin 5376
Description: The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
cnviin  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem cnviin
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5209 . 2  |-  Rel  `' |^|_
x  e.  A  B
2 r19.2z 3685 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
32expcom 425 . . . . 5  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) ) )
4 relcnv 5209 . . . . . . 7  |-  Rel  `' B
5 df-rel 4852 . . . . . . 7  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
64, 5mpbi 200 . . . . . 6  |-  `' B  C_  ( _V  X.  _V )
76a1i 11 . . . . 5  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
83, 7mprg 2743 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
9 iinss 4110 . . . 4  |-  ( E. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V ) )
108, 9syl 16 . . 3  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V )
)
11 df-rel 4852 . . 3  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1210, 11sylibr 204 . 2  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  `' B
)
13 opex 4395 . . . . 5  |-  <. b ,  a >.  e.  _V
14 eliin 4066 . . . . 5  |-  ( <.
b ,  a >.  e.  _V  ->  ( <. b ,  a >.  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  <. b ,  a >.  e.  B
) )
1513, 14ax-mp 8 . . . 4  |-  ( <.
b ,  a >.  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
16 vex 2927 . . . . 5  |-  a  e. 
_V
17 vex 2927 . . . . 5  |-  b  e. 
_V
1816, 17opelcnv 5021 . . . 4  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
19 opex 4395 . . . . . 6  |-  <. a ,  b >.  e.  _V
20 eliin 4066 . . . . . 6  |-  ( <.
a ,  b >.  e.  _V  ->  ( <. a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B ) )
2119, 20ax-mp 8 . . . . 5  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B )
2216, 17opelcnv 5021 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2322ralbii 2698 . . . . 5  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2421, 23bitri 241 . . . 4  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2515, 18, 243bitr4i 269 . . 3  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
2625eqrelriv 4936 . 2  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
271, 12, 26sylancr 645 1  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   _Vcvv 2924    C_ wss 3288   (/)c0 3596   <.cop 3785   |^|_ciin 4062    X. cxp 4843   `'ccnv 4844   Rel wrel 4850
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 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-iin 4064  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853
  Copyright terms: Public domain W3C validator