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Theorem cnvf1olem 5971
Description: Lemma for cnvf1o 5972. (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 499 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { B } )
2 1st2nd 5933 . . . . . . . 8  |-  ( ( Rel  A  /\  B  e.  A )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
32adantrr 463 . . . . . . 7  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
43sneqd 3454 . . . . . 6  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { B }  =  { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
54cnveqd 4600 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { B }  =  `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
65unieqd 3659 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { B }  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
7 1stexg 5920 . . . . . 6  |-  ( B  e.  A  ->  ( 1st `  B )  e. 
_V )
8 2ndexg 5921 . . . . . 6  |-  ( B  e.  A  ->  ( 2nd `  B )  e. 
_V )
9 opswapg 4904 . . . . . 6  |-  ( ( ( 1st `  B
)  e.  _V  /\  ( 2nd `  B )  e.  _V )  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
107, 8, 9syl2anc 403 . . . . 5  |-  ( B  e.  A  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
1110ad2antrl 474 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >. )
121, 6, 113eqtrd 2124 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  <. ( 2nd `  B ) ,  ( 1st `  B )
>. )
13 simprl 498 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  e.  A )
143, 13eqeltrrd 2165 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
15 opelcnvg 4604 . . . . . 6  |-  ( ( ( 2nd `  B
)  e.  _V  /\  ( 1st `  B )  e.  _V )  -> 
( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
168, 7, 15syl2anc 403 . . . . 5  |-  ( B  e.  A  ->  ( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
1716ad2antrl 474 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  -> 
( <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
) )
1814, 17mpbird 165 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A )
1912, 18eqeltrd 2164 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  e.  `' A
)
20 opswapg 4904 . . . . . 6  |-  ( ( ( 2nd `  B
)  e.  _V  /\  ( 1st `  B )  e.  _V )  ->  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
218, 7, 20syl2anc 403 . . . . 5  |-  ( B  e.  A  ->  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2221eqcomd 2093 . . . 4  |-  ( B  e.  A  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2322ad2antrl 474 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2412sneqd 3454 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { C }  =  { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2524cnveqd 4600 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { C }  =  `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2625unieqd 3659 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { C }  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2723, 3, 263eqtr4d 2130 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  U. `' { C } )
2819, 27jca 300 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   U.cuni 3648   `'ccnv 4427   Rel wrel 4433   ` cfv 5002   1stc1st 5891   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by:  cnvf1o  5972
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