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Mirrors > Home > ILE Home > Th. List > dmxpss2 | GIF version |
Description: Upper bound for the domain of a binary relation. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
dmxpss2 | ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmss 4841 | . 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ dom (𝐴 × 𝐵)) | |
2 | dmxpss 5074 | . 2 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | sstrdi 3182 | 1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3144 × cxp 4639 dom cdm 4641 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4647 df-dm 4651 |
This theorem is referenced by: cossxp2 5167 |
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