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Theorem cnviinm 5048
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 2176 . . 3  |-  ( y  =  a  ->  (
y  e.  A  <->  a  e.  A ) )
21cbvexv 1870 . 2  |-  ( E. y  y  e.  A  <->  E. a  a  e.  A
)
3 eleq1w 2176 . . . 4  |-  ( x  =  a  ->  (
x  e.  A  <->  a  e.  A ) )
43cbvexv 1870 . . 3  |-  ( E. x  x  e.  A  <->  E. a  a  e.  A
)
5 relcnv 4885 . . . 4  |-  Rel  `' |^|_
x  e.  A  B
6 r19.2m 3417 . . . . . . . 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 4885 . . . . . . . . 9  |-  Rel  `' B
9 df-rel 4514 . . . . . . . . 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 2464 . . . . . 6  |-  ( E. x  x  e.  A  ->  E. x  e.  A  `' B  C_  ( _V 
X.  _V ) )
13 iinss 3832 . . . . . 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 4514 . . . . 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 2661 . . . . . . . 8  |-  b  e. 
_V
18 vex 2661 . . . . . . . 8  |-  a  e. 
_V
1917, 18opex 4119 . . . . . . 7  |-  <. b ,  a >.  e.  _V
20 eliin 3786 . . . . . . 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 4689 . . . . . 6  |-  ( <.
a ,  b >.  e.  `' |^|_ x  e.  A  B 
<-> 
<. b ,  a >.  e.  |^|_ x  e.  A  B )
2318, 17opex 4119 . . . . . . . 8  |-  <. a ,  b >.  e.  _V
24 eliin 3786 . . . . . . . 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 4689 . . . . . . . 8  |-  ( <.
a ,  b >.  e.  `' B  <->  <. b ,  a
>.  e.  B )
2726ralbii 2416 . . . . . . 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 4600 . . . 4  |-  ( ( Rel  `' |^|_ x  e.  A  B  /\  Rel  |^|_ x  e.  A  `' B )  ->  `' |^|_
x  e.  A  B  =  |^|_ x  e.  A  `' B )
315, 16, 30sylancr 408 . . 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 1314   E.wex 1451    e. wcel 1463   A.wral 2391   E.wrex 2392   _Vcvv 2658    C_ wss 3039   <.cop 3498   |^|_ciin 3782    X. cxp 4505   `'ccnv 4506   Rel wrel 4512
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iin 3784  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515
This theorem is referenced by: (None)
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