MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eliniseg2 Unicode version

Theorem eliniseg2 5203
Description: Eliminate the class existence constraint in eliniseg 5192. (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 5201 . . 3  |-  Rel  `' A
2 elrelimasn 5187 . . 3  |-  ( Rel  `' A  ->  ( C  e.  ( `' A " { B } )  <-> 
B `' A C ) )
31, 2ax-mp 8 . 2  |-  ( C  e.  ( `' A " { B } )  <-> 
B `' A C )
4 relbrcnvg 5202 . 2  |-  ( Rel 
A  ->  ( B `' A C  <->  C A B ) )
53, 4syl5bb 249 1  |-  ( Rel 
A  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   {csn 3774   class class class wbr 4172   `'ccnv 4836   "cima 4840   Rel wrel 4842
This theorem is referenced by:  isunit  15717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
  Copyright terms: Public domain W3C validator