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Mirrors > Home > MPE Home > Th. List > relrelss | Structured version Visualization version GIF version |
Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Ref | Expression |
---|---|
relrelss | ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5673 | . . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V)) | |
2 | 1 | anbi2i 622 | . 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
3 | relssdmrn 6257 | . . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | |
4 | ssv 3998 | . . . . 5 ⊢ ran 𝐴 ⊆ V | |
5 | xpss12 5681 | . . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) | |
6 | 4, 5 | mpan2 688 | . . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) |
7 | 3, 6 | sylan9ss 3987 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V)) |
8 | xpss 5682 | . . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V) | |
9 | sstr 3982 | . . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V)) | |
10 | 8, 9 | mpan2 688 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V)) |
11 | df-rel 5673 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
12 | 10, 11 | sylibr 233 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴) |
13 | dmss 5892 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V)) | |
14 | vn0 4330 | . . . . . 6 ⊢ V ≠ ∅ | |
15 | dmxp 5918 | . . . . . 6 ⊢ (V ≠ ∅ → dom ((V × V) × V) = (V × V)) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ dom ((V × V) × V) = (V × V) |
17 | 13, 16 | sseqtrdi 4024 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V)) |
18 | 12, 17 | jca 511 | . . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
19 | 7, 18 | impbii 208 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V)) |
20 | 2, 19 | bitri 275 | 1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ≠ wne 2932 Vcvv 3466 ⊆ wss 3940 ∅c0 4314 × cxp 5664 dom cdm 5666 ran crn 5667 Rel wrel 5671 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-dm 5676 df-rn 5677 |
This theorem is referenced by: dftpos3 8224 tpostpos2 8227 |
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