Proof of Theorem tsrdir
| Step | Hyp | Ref
| Expression |
| 1 | | tsrps 18632 |
. . . 4
⊢ (𝐴 ∈ TosetRel → 𝐴 ∈
PosetRel) |
| 2 | | psrel 18614 |
. . . 4
⊢ (𝐴 ∈ PosetRel → Rel
𝐴) |
| 3 | 1, 2 | syl 17 |
. . 3
⊢ (𝐴 ∈ TosetRel → Rel
𝐴) |
| 4 | | psref2 18615 |
. . . . 5
⊢ (𝐴 ∈ PosetRel → (𝐴 ∩ ◡𝐴) = ( I ↾ ∪
∪ 𝐴)) |
| 5 | | inss1 4237 |
. . . . 5
⊢ (𝐴 ∩ ◡𝐴) ⊆ 𝐴 |
| 6 | 4, 5 | eqsstrrdi 4029 |
. . . 4
⊢ (𝐴 ∈ PosetRel → ( I
↾ ∪ ∪ 𝐴) ⊆ 𝐴) |
| 7 | 1, 6 | syl 17 |
. . 3
⊢ (𝐴 ∈ TosetRel → ( I
↾ ∪ ∪ 𝐴) ⊆ 𝐴) |
| 8 | 3, 7 | jca 511 |
. 2
⊢ (𝐴 ∈ TosetRel → (Rel
𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴)) |
| 9 | | pstr2 18616 |
. . . 4
⊢ (𝐴 ∈ PosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴) |
| 10 | 1, 9 | syl 17 |
. . 3
⊢ (𝐴 ∈ TosetRel → (𝐴 ∘ 𝐴) ⊆ 𝐴) |
| 11 | | psdmrn 18618 |
. . . . . . 7
⊢ (𝐴 ∈ PosetRel → (dom
𝐴 = ∪ ∪ 𝐴 ∧ ran 𝐴 = ∪ ∪ 𝐴)) |
| 12 | 1, 11 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ TosetRel → (dom
𝐴 = ∪ ∪ 𝐴 ∧ ran 𝐴 = ∪ ∪ 𝐴)) |
| 13 | 12 | simpld 494 |
. . . . 5
⊢ (𝐴 ∈ TosetRel → dom
𝐴 = ∪ ∪ 𝐴) |
| 14 | 13 | sqxpeqd 5717 |
. . . 4
⊢ (𝐴 ∈ TosetRel → (dom
𝐴 × dom 𝐴) = (∪ ∪ 𝐴 × ∪ ∪ 𝐴)) |
| 15 | | eqid 2737 |
. . . . . . 7
⊢ dom 𝐴 = dom 𝐴 |
| 16 | 15 | istsr 18628 |
. . . . . 6
⊢ (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom
𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴))) |
| 17 | 16 | simprbi 496 |
. . . . 5
⊢ (𝐴 ∈ TosetRel → (dom
𝐴 × dom 𝐴) ⊆ (𝐴 ∪ ◡𝐴)) |
| 18 | | relcoi2 6297 |
. . . . . . . 8
⊢ (Rel
𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴) |
| 19 | 3, 18 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ TosetRel → (( I
↾ ∪ ∪ 𝐴) ∘ 𝐴) = 𝐴) |
| 20 | | cnvresid 6645 |
. . . . . . . . 9
⊢ ◡( I ↾ ∪
∪ 𝐴) = ( I ↾ ∪
∪ 𝐴) |
| 21 | | cnvss 5883 |
. . . . . . . . . 10
⊢ (( I
↾ ∪ ∪ 𝐴) ⊆ 𝐴 → ◡( I ↾ ∪
∪ 𝐴) ⊆ ◡𝐴) |
| 22 | 7, 21 | syl 17 |
. . . . . . . . 9
⊢ (𝐴 ∈ TosetRel → ◡( I ↾ ∪
∪ 𝐴) ⊆ ◡𝐴) |
| 23 | 20, 22 | eqsstrrid 4023 |
. . . . . . . 8
⊢ (𝐴 ∈ TosetRel → ( I
↾ ∪ ∪ 𝐴) ⊆ ◡𝐴) |
| 24 | | coss1 5866 |
. . . . . . . 8
⊢ (( I
↾ ∪ ∪ 𝐴) ⊆ ◡𝐴 → (( I ↾ ∪ ∪ 𝐴) ∘ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)) |
| 25 | 23, 24 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ TosetRel → (( I
↾ ∪ ∪ 𝐴) ∘ 𝐴) ⊆ (◡𝐴 ∘ 𝐴)) |
| 26 | 19, 25 | eqsstrrd 4019 |
. . . . . 6
⊢ (𝐴 ∈ TosetRel → 𝐴 ⊆ (◡𝐴 ∘ 𝐴)) |
| 27 | | relcnv 6122 |
. . . . . . . 8
⊢ Rel ◡𝐴 |
| 28 | | relcoi1 6298 |
. . . . . . . 8
⊢ (Rel
◡𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴) |
| 29 | 27, 28 | ax-mp 5 |
. . . . . . 7
⊢ (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) = ◡𝐴 |
| 30 | | relcnvfld 6300 |
. . . . . . . . . . 11
⊢ (Rel
𝐴 → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴) |
| 31 | 3, 30 | syl 17 |
. . . . . . . . . 10
⊢ (𝐴 ∈ TosetRel → ∪ ∪ 𝐴 = ∪ ∪ ◡𝐴) |
| 32 | 31 | reseq2d 5997 |
. . . . . . . . 9
⊢ (𝐴 ∈ TosetRel → ( I
↾ ∪ ∪ 𝐴) = ( I ↾ ∪ ∪ ◡𝐴)) |
| 33 | 32, 7 | eqsstrrd 4019 |
. . . . . . . 8
⊢ (𝐴 ∈ TosetRel → ( I
↾ ∪ ∪ ◡𝐴) ⊆ 𝐴) |
| 34 | | coss2 5867 |
. . . . . . . 8
⊢ (( I
↾ ∪ ∪ ◡𝐴) ⊆ 𝐴 → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) ⊆ (◡𝐴 ∘ 𝐴)) |
| 35 | 33, 34 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ TosetRel → (◡𝐴 ∘ ( I ↾ ∪ ∪ ◡𝐴)) ⊆ (◡𝐴 ∘ 𝐴)) |
| 36 | 29, 35 | eqsstrrid 4023 |
. . . . . 6
⊢ (𝐴 ∈ TosetRel → ◡𝐴 ⊆ (◡𝐴 ∘ 𝐴)) |
| 37 | 26, 36 | unssd 4192 |
. . . . 5
⊢ (𝐴 ∈ TosetRel → (𝐴 ∪ ◡𝐴) ⊆ (◡𝐴 ∘ 𝐴)) |
| 38 | 17, 37 | sstrd 3994 |
. . . 4
⊢ (𝐴 ∈ TosetRel → (dom
𝐴 × dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴)) |
| 39 | 14, 38 | eqsstrrd 4019 |
. . 3
⊢ (𝐴 ∈ TosetRel → (∪ ∪ 𝐴 × ∪ ∪ 𝐴)
⊆ (◡𝐴 ∘ 𝐴)) |
| 40 | 10, 39 | jca 511 |
. 2
⊢ (𝐴 ∈ TosetRel → ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴
× ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))) |
| 41 | | eqid 2737 |
. . 3
⊢ ∪ ∪ 𝐴 = ∪ ∪ 𝐴 |
| 42 | 41 | isdir 18643 |
. 2
⊢ (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel
𝐴 ∧ ( I ↾ ∪ ∪ 𝐴) ⊆ 𝐴) ∧ ((𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (∪ ∪ 𝐴
× ∪ ∪ 𝐴) ⊆ (◡𝐴 ∘ 𝐴))))) |
| 43 | 8, 40, 42 | mpbir2and 713 |
1
⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel) |