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Mirrors > Home > ILE Home > Th. List > fconstmpt | GIF version |
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.) |
Ref | Expression |
---|---|
fconstmpt | ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 3610 | . . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
2 | 1 | anbi2i 457 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
3 | 2 | opabbii 4071 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | df-xp 4633 | . 2 ⊢ (𝐴 × {𝐵}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} | |
5 | df-mpt 4067 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 3, 4, 5 | 3eqtr4i 2208 | 1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
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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