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Theorem cnviinm 5050
Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
cnviinm  |-  ( E. y  y  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    B( x, y)

Proof of Theorem cnviinm
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1w 2178 . . 3  |-  ( y  =  a  ->  (
y  e.  A  <->  a  e.  A ) )
21cbvexv 1872 . 2  |-  ( E. y  y  e.  A  <->  E. a  a  e.  A
)
3 eleq1w 2178 . . . 4  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
43cbvexv 1872 . . 3  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
5 relcnv 4887 . . . 4  |-  Rel  `' |^|_
x  e.  A  B
6 r19.2m 3419 . . . . . . . 8  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  `' B  C_  ( _V  X.  _V ) )  ->  E. x  e.  A  `' B  C_  ( _V  X.  _V ) )
76expcom 115 . . . . . . 7  |-  ( A. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  ( E. x  x  e.  A  ->  E. x  e.  A  `' B  C_  ( _V 
X.  _V ) ) )
8 relcnv 4887 . . . . . . . . 9  |-  Rel  `' B
9 df-rel 4516 . . . . . . . . 9  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
108, 9mpbi 144 . . . . . . . 8  |-  `' B  C_  ( _V  X.  _V )
1110a1i 9 . . . . . . 7  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
127, 11mprg 2466 . . . . . 6  |-  ( E. x  x  e.  A  ->  E. x  e.  A  `' B  C_  ( _V 
X.  _V ) )
13 iinss 3834 . . . . . 6  |-  ( E. x  e.  A  `' B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  `' B  C_  ( _V  X.  _V ) )
1412, 13syl 14 . . . . 5  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
15 df-rel 4516 . . . . 5  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1614, 15sylibr 133 . . . 4  |-  ( E. x  x  e.  A  ->  Rel  |^|_ x  e.  A  `' B )
17 vex 2663 . . . . . . . 8  |-  b  e. 
_V
18 vex 2663 . . . . . . . 8  |-  a  e. 
_V
1917, 18opex 4121 . . . . . . 7  |-  <. b ,  a >.  e.  _V
20 eliin 3788 . . . . . . 7  |-  ( <.
b ,  a >.  e.  _V  ->  ( <. b ,  a >.  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  <. b ,  a >.  e.  B
) )
2119, 20ax-mp 5 . . . . . 6  |-  ( <.
b ,  a >.  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2218, 17opelcnv 4691 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
2318, 17opex 4121 . . . . . . . 8  |-  <. a ,  b >.  e.  _V
24 eliin 3788 . . . . . . . 8  |-  ( <.
a ,  b >.  e.  _V  ->  ( <. a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B ) )
2523, 24ax-mp 5 . . . . . . 7  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. a ,  b >.  e.  `' B )
2618, 17opelcnv 4691 . . . . . . . 8  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2726ralbii 2418 . . . . . . 7  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2825, 27bitri 183 . . . . . 6  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2921, 22, 283bitr4i 211 . . . . 5  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
3029eqrelriv 4602 . . . 4  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
315, 16, 30sylancr 410 . . 3  |-  ( E. x  x  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
324, 31sylbir 134 . 2  |-  ( E. a  a  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
332, 32sylbi 120 1  |-  ( E. y  y  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660    C_ wss 3041   <.cop 3500   |^|_ciin 3784    X. cxp 4507   `'ccnv 4508   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iin 3786  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by: (None)
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