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Theorem fmptsn 5914
Description: Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fmptsn  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem fmptsn
StepHypRef Expression
1 fconstmpt 4913 . 2  |-  ( { A }  X.  { B } )  =  ( x  e.  { A }  |->  B )
2 xpsng 5901 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
31, 2syl5reqr 2482 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   <.cop 3809    e. cmpt 4258    X. cxp 4868
This theorem is referenced by:  fmptap  5915  fmptapd  24051  esumsn  24446  islindf4  27240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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