Theorem fconstmpt 4822

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}) = (𝑥 ∈ 𝐴 ↦ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem fconstmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3726	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
2	1	anbi2i 461	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
3	2	opabbii 4198	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-xp 4780	. 2 ⊢ (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})}
5		df-mpt 4194	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	3, 4, 5	3eqtr4i 2269	1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {csn 3709  {copab 4191   ↦ cmpt 4192   × cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3715  df-opab 4193  df-mpt 4194  df-xp 4780

This theorem is used by:  fconst  5588  fcoconst  5879  fmptsn  5904  fconstmpo  6183  ofc12  6326  caofinvl  6328  xpexgALT  6366  inftonninf  10881  fser0const  10974  prod1dc  12355  gsumconstcmn  14168  pws0g  14215  rrgsupp  14576  psrlinv  15077  psr1clfi  15081  mpl0fi  15095  cnmptc  15385  dvexp  15814  dvexp2  15815  dvmptidcn  15817  dvmptccn  15818  dvmptid  15819  dvmptc  15820  dvmptfsum  15828  dvef  15830  elply2  15838  plyconst  15848  plycolemc  15861  nninfall  17064  nninfsellemeqinf  17071  nninfnfiinf  17078  exmidsbthrlem  17079