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Theorem cnvf1olem 6444
Description: Lemma for cnvf1o 6445. (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 734 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { B } )
2 1st2nd 6393 . . . . . . . . 9  |-  ( ( Rel  A  /\  B  e.  A )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
32adantrr 698 . . . . . . . 8  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
43sneqd 3827 . . . . . . 7  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { B }  =  { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
54cnveqd 5048 . . . . . 6  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { B }  =  `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
65unieqd 4026 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { B }  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
71, 6eqtrd 2468 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
8 opswap 5356 . . . 4  |-  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >.
97, 8syl6eq 2484 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  <. ( 2nd `  B ) ,  ( 1st `  B )
>. )
10 simprl 733 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  e.  A )
113, 10eqeltrrd 2511 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
12 fvex 5742 . . . . 5  |-  ( 2nd `  B )  e.  _V
13 fvex 5742 . . . . 5  |-  ( 1st `  B )  e.  _V
1412, 13opelcnv 5054 . . . 4  |-  ( <.
( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
1511, 14sylibr 204 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A )
169, 15eqeltrd 2510 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  e.  `' A
)
17 opswap 5356 . . . 4  |-  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >.
1817eqcomi 2440 . . 3  |-  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }
199sneqd 3827 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { C }  =  { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2019cnveqd 5048 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { C }  =  `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2120unieqd 4026 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { C }  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2218, 3, 213eqtr4a 2494 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  U. `' { C } )
2316, 22jca 519 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 359    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817   U.cuni 4015   `'ccnv 4877   Rel wrel 4883   ` cfv 5454   1stc1st 6347   2ndc2nd 6348
This theorem is referenced by:  cnvf1o  6445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-1st 6349  df-2nd 6350
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