Theorem fconstmpt 4743

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 3663	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
2	1	anbi2i 457	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
3	2	opabbii 4130	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-xp 4702	. 2 ⊢ (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})}
5		df-mpt 4126	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	3, 4, 5	3eqtr4i 2240	1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1375   ∈ wcel 2180  {csn 3646  {copab 4123   ↦ cmpt 4124   × cxp 4694

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-sn 3652  df-opab 4125  df-mpt 4126  df-xp 4702

This theorem is referenced by:  fconst  5497  fcoconst  5779  fmptsn  5801  fconstmpo  6070  ofc12  6212  caofinvl  6214  xpexgALT  6248  inftonninf  10631  fser0const  10724  prod1dc  12063  pws0g  13450  psrlinv  14613  psr1clfi  14617  mpl0fi  14631  cnmptc  14921  dvexp  15350  dvexp2  15351  dvmptidcn  15353  dvmptccn  15354  dvmptid  15355  dvmptc  15356  dvmptfsum  15364  dvef  15366  elply2  15374  plyconst  15384  plycolemc  15397  nninfall  16286  nninfsellemeqinf  16293  nninfnfiinf  16300  exmidsbthrlem  16301