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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 3544 | . . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
2 | 1 | anbi2i 452 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
3 | 2 | opabbii 3995 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | df-xp 4545 | . 2 ⊢ (𝐴 × {𝐵}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} | |
5 | df-mpt 3991 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 3, 4, 5 | 3eqtr4i 2170 | 1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3527 {copab 3988 ↦ cmpt 3989 × cxp 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sn 3533 df-opab 3990 df-mpt 3991 df-xp 4545 |
This theorem is referenced by: fconst 5318 fcoconst 5591 fmptsn 5609 fconstmpo 5866 ofc12 6002 caofinvl 6004 xpexgALT 6031 inftonninf 10214 fser0const 10289 cnmptc 12451 dvexp 12844 dvexp2 12845 dvmptidcn 12847 dvmptccn 12848 dvef 12856 nninfall 13204 nninfsellemeqinf 13212 nninffeq 13216 exmidsbthrlem 13217 |
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