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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 2740 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm2 4820 | . . 3 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
3 | opelxp1 4656 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) | |
4 | 3 | exlimiv 1598 | . . 3 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) |
5 | 2, 4 | sylbi 121 | . 2 ⊢ (𝑥 ∈ dom (𝐴 × 𝐵) → 𝑥 ∈ 𝐴) |
6 | 5 | ssriv 3159 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1492 ∈ wcel 2148 ⊆ wss 3129 〈cop 3594 × cxp 4620 dom cdm 4622 |
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 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 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 4628 df-dm 4632 |
This theorem is referenced by: rnxpss 5055 dmxpss2 5056 ssxpbm 5059 ssxp1 5060 funssxp 5380 tfrlemibfn 6322 tfr1onlembfn 6338 tfrcllembfn 6351 frecuzrdgtcl 10385 frecuzrdgdomlem 10390 dvbssntrcntop 13786 |
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