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Mirrors > Home > MPE Home > Th. List > psssdm2 | Structured version Visualization version GIF version |
Description: Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
psssdm.1 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
psssdm2 | ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmin 5744 | . . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) | |
2 | psssdm.1 | . . . . . 6 ⊢ 𝑋 = dom 𝑅 | |
3 | 2 | eqcomi 2807 | . . . . 5 ⊢ dom 𝑅 = 𝑋 |
4 | dmxpid 5764 | . . . . 5 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
5 | 3, 4 | ineq12i 4137 | . . . 4 ⊢ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴) |
6 | 1, 5 | sseqtri 3951 | . . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴) |
7 | 6 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴)) |
8 | simpr 488 | . . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴)) | |
9 | 8 | elin2d 4126 | . . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴) |
10 | elinel1 4122 | . . . . 5 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋) | |
11 | 2 | psref 17810 | . . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ 𝑋) → 𝑥𝑅𝑥) |
12 | 10, 11 | sylan2 595 | . . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥𝑅𝑥) |
13 | brinxp2 5593 | . . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥)) | |
14 | 9, 9, 12, 13 | syl21anbrc 1341 | . . 3 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥) |
15 | vex 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
16 | 15, 15 | breldm 5741 | . . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) |
17 | 14, 16 | syl 17 | . 2 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) |
18 | 7, 17 | eqelssd 3936 | 1 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 dom cdm 5519 PosetRelcps 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ps 17802 |
This theorem is referenced by: psssdm 17818 ordtrest 21807 |
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