Theorem fconstmpt 4673

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 3609	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
2	1	anbi2i 457	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
3	2	opabbii 4070	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-xp 4632	. 2 ⊢ (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})}
5		df-mpt 4066	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	3, 4, 5	3eqtr4i 2208	1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {csn 3592  {copab 4063   ↦ cmpt 4064   × cxp 4624

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 2739  df-sn 3598  df-opab 4065  df-mpt 4066  df-xp 4632

This theorem is referenced by:  fconst  5411  fcoconst  5687  fmptsn  5705  fconstmpo  5969  ofc12  6102  caofinvl  6104  xpexgALT  6133  inftonninf  10438  fser0const  10513  prod1dc  11589  cnmptc  13675  dvexp  14068  dvexp2  14069  dvmptidcn  14071  dvmptccn  14072  dvef  14081  nninfall  14640  nninfsellemeqinf  14647  exmidsbthrlem  14652