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

Theorem mptiniseg 5507
Description: Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptiniseg  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Distinct variable groups:    x, C    x, V
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem mptiniseg
StepHypRef Expression
1 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21mptpreima 5506 . 2  |-  ( `' F " { C } )  =  {
x  e.  A  |  B  e.  { C } }
3 elsnc2g 4062 . . 3  |-  ( C  e.  V  ->  ( B  e.  { C } 
<->  B  =  C ) )
43rabbidv 3101 . 2  |-  ( C  e.  V  ->  { x  e.  A  |  B  e.  { C } }  =  { x  e.  A  |  B  =  C } )
52, 4syl5eq 2510 1  |-  ( C  e.  V  ->  ( `' F " { C } )  =  {
x  e.  A  |  B  =  C }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {crab 2811   {csn 4032    |-> cmpt 4515   `'ccnv 5007   "cima 5011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-mpt 4517  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021
This theorem is referenced by:  ramub1lem1  14555  frlmsslss  18930  symgtgp  20725  csscld  21814  clsocv  21815  sqff1o  23581  dchrfi  23655  dvtanlem  30226  ftc1anclem6  30257  pwssplit4  31197  pwslnmlem2  31201
  Copyright terms: Public domain W3C validator