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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 3392 | . . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
2 | 1 | anbi2i 430 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
3 | 2 | opabbii 3824 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | df-xp 4351 | . 2 ⊢ (𝐴 × {𝐵}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})} | |
5 | df-mpt 3820 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 3, 4, 5 | 3eqtr4i 2070 | 1 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 {copab 3817 ↦ cmpt 3818 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sn 3381 df-opab 3819 df-mpt 3820 df-xp 4351 |
This theorem is referenced by: fconst 5082 fcoconst 5334 fmptsn 5352 ofc12 5731 caofinvl 5733 xpexgALT 5760 |
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