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Theorem cnviin 5391
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 5225 . 2  |-  Rel  `' |^|_
x  e.  A  B
2 r19.2z 3869 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
32expcom 441 . . . . 5  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) ) )
4 relcnv 5225 . . . . . . 7  |-  Rel  `' B
5 df-rel 4859 . . . . . . 7  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
64, 5mpbi 213 . . . . . 6  |-  `' B  C_  ( _V  X.  _V )
76a1i 11 . . . . 5  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
83, 7mprg 2762 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
9 iinss 4342 . . . 4  |-  ( E. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V ) )
108, 9syl 17 . . 3  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V )
)
11 df-rel 4859 . . 3  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1210, 11sylibr 217 . 2  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  `' B
)
13 opex 4677 . . . . 5  |-  <. b ,  a >.  e.  _V
14 eliin 4297 . . . . 5  |-  ( <.
b ,  a >.  e.  _V  ->  ( <. b ,  a >.  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  <. b ,  a >.  e.  B
) )
1513, 14ax-mp 5 . . . 4  |-  ( <.
b ,  a >.  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
16 vex 3059 . . . . 5  |-  a  e. 
_V
17 vex 3059 . . . . 5  |-  b  e. 
_V
1816, 17opelcnv 5034 . . . 4  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
19 opex 4677 . . . . . 6  |-  <. a ,  b >.  e.  _V
20 eliin 4297 . . . . . 6  |-  ( <.
a ,  b >.  e.  _V  ->  ( <. a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B ) )
2119, 20ax-mp 5 . . . . 5  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B )
2216, 17opelcnv 5034 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2322ralbii 2830 . . . . 5  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2421, 23bitri 257 . . . 4  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2515, 18, 243bitr4i 285 . . 3  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
2625eqrelriv 4946 . 2  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
271, 12, 26sylancr 674 1  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   E.wrex 2749   _Vcvv 3056    C_ wss 3415   (/)c0 3742   <.cop 3985   |^|_ciin 4292    X. cxp 4850   `'ccnv 4851   Rel wrel 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-iin 4294  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859  df-cnv 4860
This theorem is referenced by: (None)
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