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Theorem fconstmpt 4554
Description: Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
Assertion
Ref Expression
fconstmpt (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem fconstmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 velsn 3512 . . . 4 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
21anbi2i 450 . . 3 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
32opabbii 3963 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-xp 4513 . 2 (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})}
5 df-mpt 3959 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
63, 4, 53eqtr4i 2146 1 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wcel 1463  {csn 3495  {copab 3956  cmpt 3957   × cxp 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sn 3501  df-opab 3958  df-mpt 3959  df-xp 4513
This theorem is referenced by:  fconst  5286  fcoconst  5557  fmptsn  5575  fconstmpo  5832  ofc12  5968  caofinvl  5970  xpexgALT  5997  inftonninf  10154  fser0const  10229  cnmptc  12346  dvexp  12718  dvexp2  12719  dvmptidcn  12721  dvmptccn  12722  dvef  12730  nninfall  13006  nninfsellemeqinf  13014  nninffeq  13018  exmidsbthrlem  13019
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