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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 5894 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
2 | istsr.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 1, 2 | eqtr4di 2782 | . . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋) |
4 | 3 | sqxpeqd 5699 | . . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | cnveq 5864 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 5, 6 | uneq12d 4157 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅)) |
8 | 4, 7 | sseq12d 4008 | . 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
9 | df-tsr 18528 | . 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} | |
10 | 8, 9 | elrab2 3679 | 1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cun 3939 ⊆ wss 3941 × cxp 5665 ◡ccnv 5666 dom cdm 5667 PosetRelcps 18525 TosetRel ctsr 18526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-tsr 18528 |
This theorem is referenced by: istsr2 18545 tsrlemax 18547 tsrps 18548 cnvtsr 18549 letsr 18554 tsrdir 18565 |
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