ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmptsn GIF version

Theorem fmptsn 5380
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 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptsn
StepHypRef Expression
1 fconstmpt 4415 . 2 ({𝐴} × {𝐵}) = (𝑥 ∈ {𝐴} ↦ 𝐵)
2 xpsng 5366 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
31, 2syl5reqr 2103 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {csn 3403  cop 3406  cmpt 3846   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  fmptap  5381  fmptapd  5382
  Copyright terms: Public domain W3C validator