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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 5906 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
2 | istsr.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 1, 2 | eqtr4di 2786 | . . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋) |
4 | 3 | sqxpeqd 5710 | . . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | cnveq 5876 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 5, 6 | uneq12d 4163 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅)) |
8 | 4, 7 | sseq12d 4013 | . 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
9 | df-tsr 18558 | . 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} | |
10 | 8, 9 | elrab2 3685 | 1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 × cxp 5676 ◡ccnv 5677 dom cdm 5678 PosetRelcps 18555 TosetRel ctsr 18556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-cnv 5686 df-dm 5688 df-tsr 18558 |
This theorem is referenced by: istsr2 18575 tsrlemax 18577 tsrps 18578 cnvtsr 18579 letsr 18584 tsrdir 18595 |
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