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Theorem cnviin 5335
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 5169 . 2  |-  Rel  `' |^|_
x  e.  A  B
2 r19.2z 3831 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
32expcom 436 . . . . 5  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) ) )
4 relcnv 5169 . . . . . . 7  |-  Rel  `' B
5 df-rel 4803 . . . . . . 7  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
64, 5mpbi 211 . . . . . 6  |-  `' B  C_  ( _V  X.  _V )
76a1i 11 . . . . 5  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
83, 7mprg 2728 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
9 iinss 4293 . . . 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 4803 . . 3  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1210, 11sylibr 215 . 2  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  `' B
)
13 opex 4628 . . . . 5  |-  <. b ,  a >.  e.  _V
14 eliin 4248 . . . . 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 3025 . . . . 5  |-  a  e. 
_V
17 vex 3025 . . . . 5  |-  b  e. 
_V
1816, 17opelcnv 4978 . . . 4  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
19 opex 4628 . . . . . 6  |-  <. a ,  b >.  e.  _V
20 eliin 4248 . . . . . 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 4978 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2322ralbii 2796 . . . . 5  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2421, 23bitri 252 . . . 4  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2515, 18, 243bitr4i 280 . . 3  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
2625eqrelriv 4890 . 2  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
271, 12, 26sylancr 667 1  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   _Vcvv 3022    C_ wss 3379   (/)c0 3704   <.cop 3947   |^|_ciin 4243    X. cxp 4794   `'ccnv 4795   Rel wrel 4801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-iin 4245  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804
This theorem is referenced by: (None)
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