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Theorem cnvf1olem 6216
Description: Lemma for cnvf1o 6217. (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 733 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { B } )
2 1st2nd 6166 . . . . . . . . 9  |-  ( ( Rel  A  /\  B  e.  A )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
32adantrr 697 . . . . . . . 8  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
43sneqd 3653 . . . . . . 7  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { B }  =  { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
54cnveqd 4857 . . . . . 6  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { B }  =  `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
65unieqd 3838 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { B }  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
71, 6eqtrd 2315 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. } )
8 opswap 5159 . . . 4  |-  U. `' { <. ( 1st `  B
) ,  ( 2nd `  B ) >. }  =  <. ( 2nd `  B
) ,  ( 1st `  B ) >.
97, 8syl6eq 2331 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  =  <. ( 2nd `  B ) ,  ( 1st `  B )
>. )
10 simprl 732 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  e.  A )
113, 10eqeltrrd 2358 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
12 fvex 5539 . . . . 5  |-  ( 2nd `  B )  e.  _V
13 fvex 5539 . . . . 5  |-  ( 1st `  B )  e.  _V
1412, 13opelcnv 4863 . . . 4  |-  ( <.
( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A 
<-> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  A
)
1511, 14sylibr 203 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  <. ( 2nd `  B
) ,  ( 1st `  B ) >.  e.  `' A )
169, 15eqeltrd 2357 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  C  e.  `' A
)
17 opswap 5159 . . . 4  |-  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >.
1817eqcomi 2287 . . 3  |-  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. }
199sneqd 3653 . . . . 5  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  { C }  =  { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2019cnveqd 4857 . . . 4  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  `' { C }  =  `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2120unieqd 3838 . . 3  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  U. `' { C }  =  U. `' { <. ( 2nd `  B
) ,  ( 1st `  B ) >. } )
2218, 3, 213eqtr4a 2341 . 2  |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) )  ->  B  =  U. `' { C } )
2316, 22jca 518 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 358    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   U.cuni 3827   `'ccnv 4688   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  cnvf1o  6217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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