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| Mirrors > Home > MPE Home > Th. List > psssdm | Structured version Visualization version GIF version | ||
| Description: Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.) |
| Ref | Expression |
|---|---|
| psssdm.1 | ⊢ 𝑋 = dom 𝑅 |
| Ref | Expression |
|---|---|
| psssdm | ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psssdm.1 | . . 3 ⊢ 𝑋 = dom 𝑅 | |
| 2 | 1 | psssdm2 18626 | . 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) |
| 3 | sseqin2 4223 | . . 3 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
| 4 | 3 | biimpi 216 | . 2 ⊢ (𝐴 ⊆ 𝑋 → (𝑋 ∩ 𝐴) = 𝐴) |
| 5 | 2, 4 | sylan9eq 2797 | 1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 dom cdm 5685 PosetRelcps 18609 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ps 18611 |
| This theorem is referenced by: ordtrest2lem 23211 ordtrest2 23212 icopnfhmeo 24974 iccpnfhmeo 24976 xrhmeo 24977 xrge0iifhmeo 33935 |
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