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Mirrors > Home > ILE Home > Th. List > dmxpid | GIF version |
Description: The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.) |
Ref | Expression |
---|---|
dmxpid | ⊢ dom (𝐴 × 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4604 | . . 3 ⊢ (𝐴 × 𝐴) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} | |
2 | 1 | dmeqi 4799 | . 2 ⊢ dom (𝐴 × 𝐴) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} |
3 | elex2 2737 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 3 | rgen 2517 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 |
5 | dmopab3 4811 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 ↔ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴) | |
6 | 4, 5 | mpbi 144 | . 2 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴 |
7 | 2, 6 | eqtri 2185 | 1 ⊢ dom (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∀wral 2442 {copab 4036 × cxp 4596 dom cdm 4598 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-dm 4608 |
This theorem is referenced by: dmxpin 4820 xpid11 4821 sqxpeq0 5021 xpider 6563 psmetdmdm 12865 xmetdmdm 12897 |
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