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Mirrors > Home > MPE Home > Th. List > istsr | Structured version Visualization version GIF version |
Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
istsr.1 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
istsr | ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5246 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
2 | istsr.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 1, 2 | syl6eqr 2662 | . . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋) |
4 | 3 | sqxpeqd 5065 | . . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | cnveq 5218 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 5, 6 | uneq12d 3730 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅)) |
8 | 4, 7 | sseq12d 3597 | . 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
9 | df-tsr 17024 | . 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} | |
10 | 8, 9 | elrab2 3333 | 1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 × cxp 5036 ◡ccnv 5037 dom cdm 5038 PosetRelcps 17021 TosetRel ctsr 17022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-tsr 17024 |
This theorem is referenced by: istsr2 17041 tsrlemax 17043 tsrps 17044 cnvtsr 17045 letsr 17050 tsrdir 17061 |
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