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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 5542 | . . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V)) | |
2 | 1 | anbi2i 626 | . 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
3 | relssdmrn 6111 | . . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) | |
4 | ssv 3911 | . . . . 5 ⊢ ran 𝐴 ⊆ V | |
5 | xpss12 5550 | . . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) | |
6 | 4, 5 | mpan2 691 | . . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V)) |
7 | 3, 6 | sylan9ss 3900 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V)) |
8 | xpss 5551 | . . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V) | |
9 | sstr 3895 | . . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V)) | |
10 | 8, 9 | mpan2 691 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V)) |
11 | df-rel 5542 | . . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
12 | 10, 11 | sylibr 237 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴) |
13 | dmss 5755 | . . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V)) | |
14 | vn0 4237 | . . . . . 6 ⊢ V ≠ ∅ | |
15 | dmxp 5782 | . . . . . 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 3937 | . . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V)) |
18 | 12, 17 | jca 515 | . . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V))) |
19 | 7, 18 | impbii 212 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V)) |
20 | 2, 19 | bitri 278 | 1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ≠ wne 2935 Vcvv 3400 ⊆ wss 3853 ∅c0 4221 × cxp 5533 dom cdm 5535 ran crn 5536 Rel wrel 5540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-rel 5542 df-cnv 5543 df-dm 5545 df-rn 5546 |
This theorem is referenced by: dftpos3 7952 tpostpos2 7955 |
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