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Theorem relssdmrn 6114
Description: A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
relssdmrn (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (Rel 𝐴 → Rel 𝐴)
2 19.8a 2170 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3 19.8a 2170 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
4 opelxp 5584 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))
5 vex 3495 . . . . . . 7 𝑥 ∈ V
65eldm2 5763 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
7 vex 3495 . . . . . . 7 𝑦 ∈ V
87elrn2 5814 . . . . . 6 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
96, 8anbi12i 626 . . . . 5 ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
104, 9bitri 276 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
112, 3, 10sylanbrc 583 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
1211a1i 11 . 2 (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
131, 12relssdv 5654 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  wss 3933  cop 4563   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  cnvssrndm  6115  cossxp  6116  relrelss  6117  relfld  6119  fssxp  6527  oprabss  7249  cnvexg  7618  resfunexgALT  7638  cofunexg  7639  fnexALT  7641  funexw  7642  erssxp  8301  wunco  10143  trclublem  14343  trclubi  14344  trclub  14346  reltrclfv  14365  imasless  16801  sylow2a  18673  gsum2d  19021  znleval  20629  tsmsxp  22690  relfi  30280  fcnvgreu  30346  trclubNEW  39857  trrelsuperreldg  39891  trrelsuperrel2dg  39894  rp-imass  39995
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