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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 5332 | . . . 4 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) | |
2 | psssdm.1 | . . . . . 6 ⊢ 𝑋 = dom 𝑅 | |
3 | 2 | eqcomi 2631 | . . . . 5 ⊢ dom 𝑅 = 𝑋 |
4 | dmxpid 5345 | . . . . 5 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
5 | 3, 4 | ineq12i 3812 | . . . 4 ⊢ (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴) |
6 | 1, 5 | sseqtri 3637 | . . 3 ⊢ dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴) |
7 | 6 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋 ∩ 𝐴)) |
8 | inss2 3834 | . . . . . . 7 ⊢ (𝑋 ∩ 𝐴) ⊆ 𝐴 | |
9 | simpr 477 | . . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴)) | |
10 | 8, 9 | sseldi 3601 | . . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴) |
11 | inss1 3833 | . . . . . . . 8 ⊢ (𝑋 ∩ 𝐴) ⊆ 𝑋 | |
12 | 11 | sseli 3599 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋) |
13 | 2 | psref 17208 | . . . . . . 7 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ 𝑋) → 𝑥𝑅𝑥) |
14 | 12, 13 | sylan2 491 | . . . . . 6 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥𝑅𝑥) |
15 | brinxp2 5180 | . . . . . 6 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥)) | |
16 | 10, 10, 14, 15 | syl3anbrc 1246 | . . . . 5 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥) |
17 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
18 | 17, 17 | breldm 5329 | . . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) |
19 | 16, 18 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) |
20 | 19 | ex 450 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))) |
21 | 20 | ssrdv 3609 | . 2 ⊢ (𝑅 ∈ PosetRel → (𝑋 ∩ 𝐴) ⊆ dom (𝑅 ∩ (𝐴 × 𝐴))) |
22 | 7, 21 | eqssd 3620 | 1 ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 × cxp 5112 dom cdm 5114 PosetRelcps 17198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ps 17200 |
This theorem is referenced by: psssdm 17216 ordtrest 21006 |
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