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Theorem cnviin 5212
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 5051 . 2  |-  Rel  `' |^|_
x  e.  A  B
2 r19.2z 3543 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
32expcom 424 . . . . 5  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) ) )
4 relcnv 5051 . . . . . . 7  |-  Rel  `' B
5 df-rel 4696 . . . . . . 7  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
64, 5mpbi 199 . . . . . 6  |-  `' B  C_  ( _V  X.  _V )
76a1i 10 . . . . 5  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
83, 7mprg 2612 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
9 iinss 3953 . . . 4  |-  ( E. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V ) )
108, 9syl 15 . . 3  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V )
)
11 df-rel 4696 . . 3  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1210, 11sylibr 203 . 2  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  `' B
)
13 opex 4237 . . . . 5  |-  <. b ,  a >.  e.  _V
14 eliin 3910 . . . . 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 2791 . . . . 5  |-  a  e. 
_V
17 vex 2791 . . . . 5  |-  b  e. 
_V
1816, 17opelcnv 4863 . . . 4  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
19 opex 4237 . . . . . 6  |-  <. a ,  b >.  e.  _V
20 eliin 3910 . . . . . 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 4863 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2322ralbii 2567 . . . . 5  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2421, 23bitri 240 . . . 4  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2515, 18, 243bitr4i 268 . . 3  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
2625eqrelriv 4780 . 2  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
271, 12, 26sylancr 644 1  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   <.cop 3643   |^|_ciin 3906    X. cxp 4687   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  rnintintrn  25126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iin 3908  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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