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Mirrors > Home > ILE Home > Th. List > dmxpm | GIF version |
Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmxpm | ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2180 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)) | |
2 | 1 | cbvexv 1872 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵) |
3 | df-xp 4515 | . . . 4 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
4 | 3 | dmeqi 4710 | . . 3 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
5 | id 19 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵) | |
6 | 5 | ralrimivw 2483 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
7 | dmopab3 4722 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 ↔ dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴) | |
8 | 6, 7 | sylib 121 | . . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴) |
9 | 4, 8 | syl5eq 2162 | . 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴) |
10 | 2, 9 | sylbi 120 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∀wral 2393 {copab 3958 × cxp 4507 dom cdm 4509 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-dm 4519 |
This theorem is referenced by: rnxpm 4938 ssxpbm 4944 ssxp1 4945 xpexr2m 4950 relrelss 5035 unixpm 5044 exmidfodomrlemim 7025 |
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