MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relssdmrn Structured version   Visualization version   GIF version

Theorem relssdmrn 5875
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 2216 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3 19.8a 2216 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
4 opelxp 5348 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))
5 vex 3388 . . . . . . 7 𝑥 ∈ V
65eldm2 5525 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
7 vex 3388 . . . . . . 7 𝑦 ∈ V
87elrn2 5569 . . . . . 6 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
96, 8anbi12i 621 . . . . 5 ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
104, 9bitri 267 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
112, 3, 10sylanbrc 579 . . 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 5416 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wex 1875  wcel 2157  wss 3769  cop 4374   × cxp 5310  dom cdm 5312  ran crn 5313  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  cnvssrndm  5876  cossxp  5877  relrelss  5878  relfld  5880  idssxpOLD  6220  fssxp  6275  oprabss  6980  cnvexg  7347  resfunexgALT  7364  cofunexg  7365  fnexALT  7367  erssxp  8005  wunco  9843  trclublem  14077  trclubi  14078  trclub  14080  reltrclfv  14099  imasless  16515  sylow2a  18347  gsum2d  18686  znleval  20224  tsmsxp  22286  relfi  29932  fcnvgreu  29990  rtrclex  38707  trclubNEW  38709  rtrclexi  38711  trrelsuperreldg  38743  trrelsuperrel2dg  38746  rp-imass  38847  idhe  38863
  Copyright terms: Public domain W3C validator