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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 2141 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)) | |
2 | 1 | cbvexv 1836 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵) |
3 | df-xp 4369 | . . . 4 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
4 | 3 | dmeqi 4554 | . . 3 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
5 | id 19 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵) | |
6 | 5 | ralrimivw 2435 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
7 | dmopab3 4566 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵 ↔ dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴) | |
8 | 6, 7 | sylib 120 | . . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} = 𝐴) |
9 | 4, 8 | syl5eq 2125 | . 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴) |
10 | 2, 9 | sylbi 119 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∀wral 2348 {copab 3838 × cxp 4361 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-dm 4373 |
This theorem is referenced by: dmxpinm 4574 xpid11m 4575 rnxpm 4772 ssxpbm 4776 ssxp1 4777 xpexr2m 4782 relrelss 4864 unixpm 4873 xpiderm 6200 |
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