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Theorem relrelss 5697
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 5150 . . 3 (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
21anbi2i 730 . 2 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3 relssdmrn 5694 . . . 4 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4 ssv 3658 . . . . 5 ran 𝐴 ⊆ V
5 xpss12 5158 . . . . 5 ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
64, 5mpan2 707 . . . 4 (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
73, 6sylan9ss 3649 . . 3 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8 xpss 5159 . . . . . 6 ((V × V) × V) ⊆ (V × V)
9 sstr 3644 . . . . . 6 ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
108, 9mpan2 707 . . . . 5 (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11 df-rel 5150 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
1210, 11sylibr 224 . . . 4 (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13 dmss 5355 . . . . 5 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14 vn0 3957 . . . . . 6 V ≠ ∅
15 dmxp 5376 . . . . . 6 (V ≠ ∅ → dom ((V × V) × V) = (V × V))
1614, 15ax-mp 5 . . . . 5 dom ((V × V) × V) = (V × V)
1713, 16syl6sseq 3684 . . . 4 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
1812, 17jca 553 . . 3 (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
197, 18impbii 199 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
202, 19bitri 264 1 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wne 2823  Vcvv 3231  wss 3607  c0 3948   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  dftpos3  7415  tpostpos2  7418
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