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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 5562 | . . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V)) | |
2 | 1 | anbi2i 624 | . 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
3 | relssdmrn 6121 | . . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | |
4 | ssv 3991 | . . . . 5 ⊢ ran 𝐴 ⊆ V | |
5 | xpss12 5570 | . . . . 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 3980 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V)) |
8 | xpss 5571 | . . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V) | |
9 | sstr 3975 | . . . . . 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 5562 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
12 | 10, 11 | sylibr 236 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴) |
13 | dmss 5771 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V)) | |
14 | vn0 4304 | . . . . . 6 ⊢ V ≠ ∅ | |
15 | dmxp 5799 | . . . . . 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 4017 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V)) |
18 | 12, 17 | jca 514 | . . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
19 | 7, 18 | impbii 211 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V)) |
20 | 2, 19 | bitri 277 | 1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 × cxp 5553 dom cdm 5555 ran crn 5556 Rel wrel 5560 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: dftpos3 7910 tpostpos2 7913 |
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