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Theorem cnvf1olem 5873
Description: Lemma for cnvf1o 5874. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
cnvf1olem  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  -> 
( C  e.  `' A  /\  B  =  U. `' { C } ) )

Proof of Theorem cnvf1olem
StepHypRef Expression
1 simprr 492 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { B } )
2 1st2nd 5835 . . . . . . . 8  |-  ( ( Rel  A  /\  B  e.  A )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
32adantrr 456 . . . . . . 7  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
43sneqd 3416 . . . . . 6  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { B }  =  { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
54cnveqd 4539 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { B }  =  `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
65unieqd 3619 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { B }  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
7 1stexg 5822 . . . . . 6  |-  ( B  e.  A  ->  ( 1st `  B )  e. 
_V )
8 2ndexg 5823 . . . . . 6  |-  ( B  e.  A  ->  ( 2nd `  B )  e. 
_V )
9 opswapg 4835 . . . . . 6  |-  ( ( ( 1st `  B
)  e.  _V  /\  ( 2nd `  B )  e.  _V )  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
107, 8, 9syl2anc 397 . . . . 5  |-  ( B  e.  A  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
1110ad2antrl 467 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
121, 6, 113eqtrd 2092 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  <. ( 2nd `  B ) ,  ( 1st `  B )
>. )
13 simprl 491 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  e.  A )
143, 13eqeltrrd 2131 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
15 opelcnvg 4543 . . . . . 6  |-  ( ( ( 2nd `  B
)  e.  _V  /\  ( 1st `  B )  e.  _V )  -> 
( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
168, 7, 15syl2anc 397 . . . . 5  |-  ( B  e.  A  ->  ( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
1716ad2antrl 467 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  -> 
( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
1814, 17mpbird 160 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A )
1912, 18eqeltrd 2130 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  e.  `' A
)
20 opswapg 4835 . . . . . 6  |-  ( ( ( 2nd `  B
)  e.  _V  /\  ( 1st `  B )  e.  _V )  ->  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
218, 7, 20syl2anc 397 . . . . 5  |-  ( B  e.  A  ->  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2221eqcomd 2061 . . . 4  |-  ( B  e.  A  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2322ad2antrl 467 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2412sneqd 3416 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { C }  =  { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2524cnveqd 4539 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { C }  =  `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2625unieqd 3619 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { C }  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2723, 3, 263eqtr4d 2098 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  U. `' { C } )
2819, 27jca 294 1  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  -> 
( C  e.  `' A  /\  B  =  U. `' { C } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403   <.cop 3406   U.cuni 3608   `'ccnv 4372   Rel wrel 4378   ` cfv 4930   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  cnvf1o  5874
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