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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 5689 | . . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V)) | |
2 | 1 | anbi2i 621 | . 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
3 | relssdmrn 6277 | . . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | |
4 | ssv 4006 | . . . . 5 ⊢ ran 𝐴 ⊆ V | |
5 | xpss12 5697 | . . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) | |
6 | 4, 5 | mpan2 689 | . . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) |
7 | 3, 6 | sylan9ss 3995 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V)) |
8 | xpss 5698 | . . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V) | |
9 | sstr 3990 | . . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V)) | |
10 | 8, 9 | mpan2 689 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V)) |
11 | df-rel 5689 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
12 | 10, 11 | sylibr 233 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴) |
13 | dmss 5909 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V)) | |
14 | vn0 4342 | . . . . . 6 ⊢ V ≠ ∅ | |
15 | dmxp 5935 | . . . . . 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 4032 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V)) |
18 | 12, 17 | jca 510 | . . 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 274 | 1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ≠ wne 2937 Vcvv 3473 ⊆ wss 3949 ∅c0 4326 × cxp 5680 dom cdm 5682 ran crn 5683 Rel wrel 5687 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: dftpos3 8258 tpostpos2 8261 |
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