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Theorem cnviinm 5169
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 2238 . . 3  |-  ( y  =  a  ->  (
y  e.  A  <->  a  e.  A ) )
21cbvexv 1918 . 2  |-  ( E. y  y  e.  A  <->  E. a  a  e.  A
)
3 eleq1w 2238 . . . 4  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
43cbvexv 1918 . . 3  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
5 relcnv 5005 . . . 4  |-  Rel  `' |^|_
x  e.  A  B
6 r19.2m 3509 . . . . . . . 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 116 . . . . . . 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 5005 . . . . . . . . 9  |-  Rel  `' B
9 df-rel 4632 . . . . . . . . 9  |-  ( Rel  `' B  <->  `' B  C_  ( _V 
X.  _V ) )
108, 9mpbi 145 . . . . . . . 8  |-  `' B  C_  ( _V  X.  _V )
1110a1i 9 . . . . . . 7  |-  ( x  e.  A  ->  `' B  C_  ( _V  X.  _V ) )
127, 11mprg 2534 . . . . . 6  |-  ( E. x  x  e.  A  ->  E. x  e.  A  `' B  C_  ( _V 
X.  _V ) )
13 iinss 3937 . . . . . 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 4632 . . . . 5  |-  ( Rel  |^|_ x  e.  A  `' B 
<-> 
|^|_ x  e.  A  `' B  C_  ( _V 
X.  _V ) )
1614, 15sylibr 134 . . . 4  |-  ( E. x  x  e.  A  ->  Rel  |^|_ x  e.  A  `' B )
17 vex 2740 . . . . . . . 8  |-  b  e. 
_V
18 vex 2740 . . . . . . . 8  |-  a  e. 
_V
1917, 18opex 4228 . . . . . . 7  |-  <. b ,  a >.  e.  _V
20 eliin 3891 . . . . . . 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 4808 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
2318, 17opex 4228 . . . . . . . 8  |-  <. a ,  b >.  e.  _V
24 eliin 3891 . . . . . . . 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 4808 . . . . . . . 8  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2726ralbii 2483 . . . . . . 7  |-  ( A. x  e.  A  <. a ,  b >.  e.  `' B 
<-> 
A. x  e.  A  <. b ,  a >.  e.  B )
2825, 27bitri 184 . . . . . 6  |-  ( <.
a ,  b >.  e.  |^|_ x  e.  A  `' B  <->  A. x  e.  A  <. b ,  a >.  e.  B )
2921, 22, 283bitr4i 212 . . . . 5  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. a ,  b >.  e.  |^|_ x  e.  A  `' B )
3029eqrelriv 4718 . . . 4  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
315, 16, 30sylancr 414 . . 3  |-  ( E. x  x  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
324, 31sylbir 135 . 2  |-  ( E. a  a  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_ x  e.  A  `' B )
332, 32sylbi 121 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 105    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737    C_ wss 3129   <.cop 3595   |^|_ciin 3887    X. cxp 4623   `'ccnv 4624   Rel wrel 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-iin 3889  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633
This theorem is referenced by: (None)
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