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

Theorem elimasng 5055
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
Assertion
Ref Expression
elimasng  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )

Proof of Theorem elimasng
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3664 . . . . 5  |-  ( y  =  B  ->  { y }  =  { B } )
21imaeq2d 5028 . . . 4  |-  ( y  =  B  ->  ( A " { y } )  =  ( A
" { B }
) )
32eleq2d 2363 . . 3  |-  ( y  =  B  ->  (
z  e.  ( A
" { y } )  <->  z  e.  ( A " { B } ) ) )
4 opeq1 3812 . . . 4  |-  ( y  =  B  ->  <. y ,  z >.  =  <. B ,  z >. )
54eleq1d 2362 . . 3  |-  ( y  =  B  ->  ( <. y ,  z >.  e.  A  <->  <. B ,  z
>.  e.  A ) )
63, 5bibi12d 312 . 2  |-  ( y  =  B  ->  (
( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A
)  <->  ( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A ) ) )
7 eleq1 2356 . . 3  |-  ( z  =  C  ->  (
z  e.  ( A
" { B }
)  <->  C  e.  ( A " { B }
) ) )
8 opeq2 3813 . . . 4  |-  ( z  =  C  ->  <. B , 
z >.  =  <. B ,  C >. )
98eleq1d 2362 . . 3  |-  ( z  =  C  ->  ( <. B ,  z >.  e.  A  <->  <. B ,  C >.  e.  A ) )
107, 9bibi12d 312 . 2  |-  ( z  =  C  ->  (
( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A )  <-> 
( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) ) )
11 vex 2804 . . 3  |-  y  e. 
_V
12 vex 2804 . . 3  |-  z  e. 
_V
1311, 12elimasn 5054 . 2  |-  ( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A )
146, 10, 13vtocl2g 2860 1  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   "cima 4708
This theorem is referenced by:  elimasni  5056  eliniseg  5058  dffv3  5537  fvimacnv  5656  elecg  6714  elpredim  24247  elpredg  24249  funpartfv  24555  eltail  26426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
  Copyright terms: Public domain W3C validator