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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 5774 | . . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅) | |
2 | istsr.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 1, 2 | syl6eqr 2876 | . . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋) |
4 | 3 | sqxpeqd 5589 | . . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | cnveq 5746 | . . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
7 | 5, 6 | uneq12d 4142 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅)) |
8 | 4, 7 | sseq12d 4002 | . 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅))) |
9 | df-tsr 17813 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 × cxp 5555 ◡ccnv 5556 dom cdm 5557 PosetRelcps 17810 TosetRel ctsr 17811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-tsr 17813 |
This theorem is referenced by: istsr2 17830 tsrlemax 17832 tsrps 17833 cnvtsr 17834 letsr 17839 tsrdir 17850 |
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