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Theorem elcnvlem 36278
Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
Hypothesis
Ref Expression
elcnvlem.f  |-  F  =  ( x  e.  ( _V  X.  _V )  |-> 
<. ( 2nd `  x
) ,  ( 1st `  x ) >. )
Assertion
Ref Expression
elcnvlem  |-  ( A  e.  `' B  <->  ( A  e.  ( _V  X.  _V )  /\  ( F `  A )  e.  B
) )

Proof of Theorem elcnvlem
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnv2 5017 . 2  |-  ( A  e.  `' B  <->  E. u E. v ( A  = 
<. u ,  v >.  /\  <. v ,  u >.  e.  B ) )
2 fveq2 5879 . . . . 5  |-  ( A  =  <. u ,  v
>.  ->  ( F `  A )  =  ( F `  <. u ,  v >. )
)
3 vex 3034 . . . . . . 7  |-  u  e. 
_V
4 vex 3034 . . . . . . 7  |-  v  e. 
_V
53, 4opelvv 4886 . . . . . 6  |-  <. u ,  v >.  e.  ( _V  X.  _V )
63, 4op2ndd 6823 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( 2nd `  x
)  =  v )
73, 4op1std 6822 . . . . . . . 8  |-  ( x  =  <. u ,  v
>.  ->  ( 1st `  x
)  =  u )
86, 7opeq12d 4166 . . . . . . 7  |-  ( x  =  <. u ,  v
>.  ->  <. ( 2nd `  x
) ,  ( 1st `  x ) >.  =  <. v ,  u >. )
9 elcnvlem.f . . . . . . 7  |-  F  =  ( x  e.  ( _V  X.  _V )  |-> 
<. ( 2nd `  x
) ,  ( 1st `  x ) >. )
10 opex 4664 . . . . . . 7  |-  <. v ,  u >.  e.  _V
118, 9, 10fvmpt 5963 . . . . . 6  |-  ( <.
u ,  v >.  e.  ( _V  X.  _V )  ->  ( F `  <. u ,  v >.
)  =  <. v ,  u >. )
125, 11ax-mp 5 . . . . 5  |-  ( F `
 <. u ,  v
>. )  =  <. v ,  u >.
132, 12syl6eq 2521 . . . 4  |-  ( A  =  <. u ,  v
>.  ->  ( F `  A )  =  <. v ,  u >. )
1413eleq1d 2533 . . 3  |-  ( A  =  <. u ,  v
>.  ->  ( ( F `
 A )  e.  B  <->  <. v ,  u >.  e.  B ) )
1514copsex2gb 4950 . 2  |-  ( E. u E. v ( A  =  <. u ,  v >.  /\  <. v ,  u >.  e.  B
)  <->  ( A  e.  ( _V  X.  _V )  /\  ( F `  A )  e.  B
) )
161, 15bitri 257 1  |-  ( A  e.  `' B  <->  ( A  e.  ( _V  X.  _V )  /\  ( F `  A )  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   <.cop 3965    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   ` cfv 5589   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813
This theorem is referenced by:  elcnvintab  36279
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