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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 4629 | . . 3 ⊢ (𝐴 × 𝐴) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} | |
2 | 1 | dmeqi 4824 | . 2 ⊢ dom (𝐴 × 𝐴) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} |
3 | elex2 2753 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) | |
4 | 3 | rgen 2530 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 |
5 | dmopab3 4836 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐴 ↔ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴) | |
6 | 4, 5 | mpbi 145 | . 2 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)} = 𝐴 |
7 | 2, 6 | eqtri 2198 | 1 ⊢ dom (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 {copab 4060 × cxp 4621 dom cdm 4623 |
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-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-dm 4633 |
This theorem is referenced by: dmxpin 4845 xpid11 4846 sqxpeq0 5048 xpider 6600 psmetdmdm 13484 xmetdmdm 13516 |
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