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Theorem relrelss 5845
Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
relrelss ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Proof of Theorem relrelss
StepHypRef Expression
1 df-rel 5284 . . 3 (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
21anbi2i 616 . 2 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3 relssdmrn 5842 . . . 4 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4 ssv 3785 . . . . 5 ran 𝐴 ⊆ V
5 xpss12 5292 . . . . 5 ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
64, 5mpan2 682 . . . 4 (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
73, 6sylan9ss 3774 . . 3 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8 xpss 5293 . . . . . 6 ((V × V) × V) ⊆ (V × V)
9 sstr 3769 . . . . . 6 ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
108, 9mpan2 682 . . . . 5 (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11 df-rel 5284 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
1210, 11sylibr 225 . . . 4 (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13 dmss 5491 . . . . 5 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14 vn0 4089 . . . . . 6 V ≠ ∅
15 dmxp 5512 . . . . . 6 (V ≠ ∅ → dom ((V × V) × V) = (V × V))
1614, 15ax-mp 5 . . . . 5 dom ((V × V) × V) = (V × V)
1713, 16syl6sseq 3811 . . . 4 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
1812, 17jca 507 . . 3 (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
197, 18impbii 200 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
202, 19bitri 266 1 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wne 2937  Vcvv 3350  wss 3732  c0 4079   × cxp 5275  dom cdm 5277  ran crn 5278  Rel wrel 5282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288
This theorem is referenced by:  dftpos3  7573  tpostpos2  7576
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