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Theorem relssdmrn 5573
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 2039 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3 19.8a 2039 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
4 opelxp 5070 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))
5 vex 3176 . . . . . . 7 𝑥 ∈ V
65eldm2 5244 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
7 vex 3176 . . . . . . 7 𝑦 ∈ V
87elrn2 5286 . . . . . 6 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
96, 8anbi12i 729 . . . . 5 ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
104, 9bitri 263 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
112, 3, 10sylanbrc 695 . . 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 5135 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1695  wcel 1977  wss 3540  cop 4131   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049
This theorem is referenced by:  cnvssrndm  5574  cossxp  5575  relrelss  5576  relfld  5578  fssxp  5973  oprabss  6644  cnvexg  7005  resfunexgALT  7022  cofunexg  7023  fnexALT  7025  erssxp  7652  wunco  9434  trclublem  13582  trclubi  13583  trclubiOLD  13584  trclub  13587  reltrclfv  13606  imasless  16023  sylow2a  17857  gsum2d  18194  znleval  19722  tsmsxp  21768  relfi  28797  idssxp  28811  fcnvgreu  28855  rtrclex  36943  trclubNEW  36945  rtrclexi  36947  trrelsuperreldg  36979  trrelsuperrel2dg  36982  rp-imass  37085  idhe  37101
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