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Theorem cnvf1olem 6879
Description: Lemma for cnvf1o 6880. (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 756 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { B } )
2 1st2nd 6827 . . . . . . . . 9  |-  ( ( Rel  A  /\  B  e.  A )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
32adantrr 716 . . . . . . . 8  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
43sneqd 4022 . . . . . . 7  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { B }  =  { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
54cnveqd 5164 . . . . . 6  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { B }  =  `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
65unieqd 4240 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { B }  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
71, 6eqtrd 2482 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
8 opswap 5481 . . . 4  |-  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >.
97, 8syl6eq 2498 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  <. ( 2nd `  B ) ,  ( 1st `  B )
>. )
10 simprl 755 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  e.  A )
113, 10eqeltrrd 2530 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
12 fvex 5862 . . . . 5  |-  ( 2nd `  B )  e.  _V
13 fvex 5862 . . . . 5  |-  ( 1st `  B )  e.  _V
1412, 13opelcnv 5170 . . . 4  |-  ( <.
( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
1511, 14sylibr 212 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A )
169, 15eqeltrd 2529 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  e.  `' A
)
17 opswap 5481 . . . 4  |-  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >.
1817eqcomi 2454 . . 3  |-  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }
199sneqd 4022 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { C }  =  { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2019cnveqd 5164 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { C }  =  `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2120unieqd 4240 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { C }  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2218, 3, 213eqtr4a 2508 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  U. `' { C } )
2316, 22jca 532 1  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  -> 
( C  e.  `' A  /\  B  =  U. `' { C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   {csn 4010   <.cop 4016   U.cuni 4230   `'ccnv 4984   Rel wrel 4990   ` cfv 5574   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fv 5582  df-1st 6781  df-2nd 6782
This theorem is referenced by:  cnvf1o  6880  fcnvgreu  27379
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