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Theorem relrelss 6262
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 5673 . . 3 (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
21anbi2i 622 . 2 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3 relssdmrn 6257 . . . 4 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4 ssv 3998 . . . . 5 ran 𝐴 ⊆ V
5 xpss12 5681 . . . . 5 ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
64, 5mpan2 688 . . . 4 (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
73, 6sylan9ss 3987 . . 3 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8 xpss 5682 . . . . . 6 ((V × V) × V) ⊆ (V × V)
9 sstr 3982 . . . . . 6 ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
108, 9mpan2 688 . . . . 5 (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11 df-rel 5673 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
1210, 11sylibr 233 . . . 4 (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13 dmss 5892 . . . . 5 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14 vn0 4330 . . . . . 6 V ≠ ∅
15 dmxp 5918 . . . . . 6 (V ≠ ∅ → dom ((V × V) × V) = (V × V))
1614, 15ax-mp 5 . . . . 5 dom ((V × V) × V) = (V × V)
1713, 16sseqtrdi 4024 . . . 4 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
1812, 17jca 511 . . 3 (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
197, 18impbii 208 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
202, 19bitri 275 1 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wne 2932  Vcvv 3466  wss 3940  c0 4314   × cxp 5664  dom cdm 5666  ran crn 5667  Rel wrel 5671
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by:  dftpos3  8224  tpostpos2  8227
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