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Theorem relrelss 5021
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 4504 . . 3 (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
21anbi2i 450 . 2 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3 relssdmrn 5015 . . . 4 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4 ssv 3083 . . . . 5 ran 𝐴 ⊆ V
5 xpss12 4604 . . . . 5 ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
64, 5mpan2 419 . . . 4 (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
73, 6sylan9ss 3074 . . 3 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8 xpss 4605 . . . . . 6 ((V × V) × V) ⊆ (V × V)
9 sstr 3069 . . . . . 6 ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
108, 9mpan2 419 . . . . 5 (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11 df-rel 4504 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
1210, 11sylibr 133 . . . 4 (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13 dmss 4696 . . . . 5 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14 vn0m 3338 . . . . . 6 𝑥 𝑥 ∈ V
15 dmxpm 4717 . . . . . 6 (∃𝑥 𝑥 ∈ V → dom ((V × V) × V) = (V × V))
1614, 15ax-mp 7 . . . . 5 dom ((V × V) × V) = (V × V)
1713, 16syl6sseq 3109 . . . 4 (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
1812, 17jca 302 . . 3 (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
197, 18impbii 125 . 2 ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
202, 19bitri 183 1 ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1312  wex 1449  wcel 1461  Vcvv 2655  wss 3035   × cxp 4495  dom cdm 4497  ran crn 4498  Rel wrel 4502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508
This theorem is referenced by:  dftpos3  6110  tpostpos2  6113
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