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Theorem fconstmpt 4674
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 3610 . . . 4 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
21anbi2i 457 . . 3 ((𝑥𝐴𝑦 ∈ {𝐵}) ↔ (𝑥𝐴𝑦 = 𝐵))
32opabbii 4071 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
4 df-xp 4633 . 2 (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ {𝐵})}
5 df-mpt 4067 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
63, 4, 53eqtr4i 2208 1 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  {csn 3593  {copab 4064  cmpt 4065   × cxp 4625
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-sn 3599  df-opab 4066  df-mpt 4067  df-xp 4633
This theorem is referenced by:  fconst  5412  fcoconst  5688  fmptsn  5706  fconstmpo  5970  ofc12  6103  caofinvl  6105  xpexgALT  6134  inftonninf  10441  fser0const  10516  prod1dc  11594  cnmptc  13785  dvexp  14178  dvexp2  14179  dvmptidcn  14181  dvmptccn  14182  dvef  14191  nninfall  14761  nninfsellemeqinf  14768  exmidsbthrlem  14773
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