Theorem fmptsn 5873

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

Step	Hyp	Ref	Expression
1		xpsng 5853	. 2 |- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
2		fconstmpt 4797	. 2 |- ( { A } X. { B } ) = ( x e. { A } |-> B )
3	1, 2	eqtr3di 2280	1 |- ( ( A e. V /\ B e. W ) -> { <. A , B >. } = ( x e. { A } |-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {csn 3689   <.cop 3692   |-> cmpt 4171   X. cxp 4747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by:  fmptap  5874  fmptapd  5875  gsumsplit0  14063  gfsumsn  16867