Theorem fconstmpt 4711

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 3640	. . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
2	1	anbi2i 457	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
3	2	opabbii 4101	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4		df-xp 4670	. 2 ⊢ (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})}
5		df-mpt 4097	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	3, 4, 5	3eqtr4i 2227	1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {csn 3623  {copab 4094   ↦ cmpt 4095   × cxp 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3629  df-opab 4096  df-mpt 4097  df-xp 4670

This theorem is referenced by:  fconst  5456  fcoconst  5736  fmptsn  5754  fconstmpo  6021  ofc12  6163  caofinvl  6165  xpexgALT  6199  inftonninf  10553  fser0const  10646  prod1dc  11770  pws0g  13155  psrlinv  14314  psr1clfi  14318  cnmptc  14604  dvexp  15033  dvexp2  15034  dvmptidcn  15036  dvmptccn  15037  dvmptid  15038  dvmptc  15039  dvmptfsum  15047  dvef  15049  elply2  15057  plyconst  15067  plycolemc  15080  nninfall  15742  nninfsellemeqinf  15749  nninfnfiinf  15756  exmidsbthrlem  15757