Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  intimasn2 Structured version   Unicode version

Theorem intimasn2 36221
Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimasn2  |-  ( B  e.  V  ->  ( |^| A " { B } )  =  |^|_ x  e.  A  ( x
" { B }
) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem intimasn2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 intimasn 36220 . 2  |-  ( B  e.  V  ->  ( |^| A " { B } )  =  |^| { y  |  E. x  e.  A  y  =  ( x " { B } ) } )
2 intima0 36210 . 2  |-  |^|_ x  e.  A  ( x " { B } )  =  |^| { y  |  E. x  e.  A  y  =  ( x " { B } ) }
31, 2syl6eqr 2481 1  |-  ( B  e.  V  ->  ( |^| A " { B } )  =  |^|_ x  e.  A  ( x
" { B }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   {cab 2407   E.wrex 2772   {csn 3998   |^|cint 4255   |^|_ciin 4300   "cima 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-iin 4302  df-br 4424  df-opab 4483  df-xp 4859  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator