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

Theorem eliniseg2 5132
Description: Eliminate the class existence constraint in eliniseg 5121. (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 5130 . . 3  |-  Rel  `' A
2 elrelimasn 5116 . . 3  |-  ( Rel  `' A  ->  ( C  e.  ( `' A " { B } )  <-> 
B `' A C ) )
31, 2ax-mp 8 . 2  |-  ( C  e.  ( `' A " { B } )  <-> 
B `' A C )
4 relbrcnvg 5131 . 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 1710   {csn 3716   class class class wbr 4102   `'ccnv 4767   "cima 4771   Rel wrel 4773
This theorem is referenced by:  isunit  15532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781
  Copyright terms: Public domain W3C validator