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Theorem cnviin 5412
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 5245 . 2  |-  Rel  `' |^|_
x  e.  A  B
2 r19.2z 3719 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
32expcom 426 . . . . 5  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) ) )
4 relcnv 5245 . . . . . . 7  |-  Rel  `' B
5 df-rel 4888 . . . . . . 7  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
64, 5mpbi 201 . . . . . 6  |-  `' B  C_  ( _V  X.  _V )
76a1i 11 . . . . 5  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
83, 7mprg 2777 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
9 iinss 4144 . . . 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 4888 . . 3  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1210, 11sylibr 205 . 2  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  `' B
)
13 opex 4430 . . . . 5  |-  <. b ,  a >.  e.  _V
14 eliin 4100 . . . . 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 2961 . . . . 5  |-  a  e. 
_V
17 vex 2961 . . . . 5  |-  b  e. 
_V
1816, 17opelcnv 5057 . . . 4  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
19 opex 4430 . . . . . 6  |-  <. a ,  b >.  e.  _V
20 eliin 4100 . . . . . 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 5057 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2322ralbii 2731 . . . . 5  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2421, 23bitri 242 . . . 4  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2515, 18, 243bitr4i 270 . . 3  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
2625eqrelriv 4972 . 2  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
271, 12, 26sylancr 646 1  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   (/)c0 3630   <.cop 3819   |^|_ciin 4096    X. cxp 4879   `'ccnv 4880   Rel wrel 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-iin 4098  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889
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