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Mirrors > Home > ILE Home > Th. List > dmxpss | GIF version |
Description: The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.) |
Ref | Expression |
---|---|
dmxpss | ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4533 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
3 | opelxp1 4377 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) | |
4 | 3 | exlimiv 1489 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) |
5 | 2, 4 | sylbi 114 | . 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) |
6 | 5 | ssriv 2949 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1381 ∈ wcel 1393 ⊆ wss 2917 〈cop 3378 × cxp 4343 dom cdm 4345 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-dm 4355 |
This theorem is referenced by: rnxpss 4754 ssxpbm 4756 ssxp1 4757 funssxp 5060 tfrlemibfn 5942 frecuzrdgfn 9198 |
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