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

Theorem relssdmrn 5800
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 2205 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3 19.8a 2205 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
4 opelxp 5286 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))
5 vex 3352 . . . . . . 7 𝑥 ∈ V
65eldm2 5460 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
7 vex 3352 . . . . . . 7 𝑦 ∈ V
87elrn2 5503 . . . . . 6 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
96, 8anbi12i 604 . . . . 5 ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
104, 9bitri 264 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
112, 3, 10sylanbrc 564 . . 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 5352 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1851  wcel 2144  wss 3721  cop 4320   × cxp 5247  dom cdm 5249  ran crn 5250  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by:  cnvssrndm  5801  cossxp  5802  relrelss  5803  relfld  5805  idssxp  6149  fssxp  6200  oprabss  6892  cnvexg  7258  resfunexgALT  7275  cofunexg  7276  fnexALT  7278  erssxp  7918  wunco  9756  trclublem  13943  trclubi  13944  trclub  13946  reltrclfv  13965  imasless  16407  sylow2a  18240  gsum2d  18577  znleval  20117  tsmsxp  22177  relfi  29747  fcnvgreu  29806  rtrclex  38443  trclubNEW  38445  rtrclexi  38447  trrelsuperreldg  38479  trrelsuperrel2dg  38482  rp-imass  38584  idhe  38600
  Copyright terms: Public domain W3C validator