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Theorem fmptsn 6092
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 5049 . 2  |-  ( { A }  X.  { B } )  =  ( x  e.  { A }  |->  B )
2 xpsng 6073 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B } )  =  { <. A ,  B >. } )
31, 2syl5reqr 2523 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033   <.cop 4039    |-> cmpt 4511    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601
This theorem is referenced by:  fmptap  6095  fmptapd  6096  islindf4  18742  mat1dimscm  18846  istrkg2ld  23724  esumsn  27888  ofs1  28315  ofcs1  28316  signstf0  28341  ptrest  29966  fmptsnxp  31336
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