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Theorem eliniseg2 5053
Description: Eliminate the class existence constraint in eliniseg 5042. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
Assertion
Ref Expression
eliniseg2  |-  ( Rel 
A  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )

Proof of Theorem eliniseg2
StepHypRef Expression
1 relcnv 5051 . . 3  |-  Rel  `' A
2 elrelimasn 5037 . . 3  |-  ( Rel  `' A  ->  ( C  e.  ( `' A " { B } )  <-> 
B `' A C ) )
31, 2ax-mp 8 . 2  |-  ( C  e.  ( `' A " { B } )  <-> 
B `' A C )
4 relbrcnvg 5052 . 2  |-  ( Rel 
A  ->  ( B `' A C  <->  C A B ) )
53, 4syl5bb 248 1  |-  ( Rel 
A  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   {csn 3640   class class class wbr 4023   `'ccnv 4688   "cima 4692   Rel wrel 4694
This theorem is referenced by:  isunit  15439
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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