Theorem relssdmrn 6270

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.) (Proof shortened by SN, 23-Dec-2024.)

Assertion

Ref	Expression
relssdmrn	⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5896	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5932	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5699	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
7	6	a1i 11	. 2 ⊢ (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
8	1, 7	relssdv 5773	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ⊆ wss 3904  ⟨cop 4594   × cxp 5658  dom cdm 5660  ran crn 5661  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by:  resssxp  6271  cnvssrndm  6272  cossxp  6273  relrelss  6274  relfld  6276  fssxp  6733  oprabss  7520  cnvexg  7919  resfunexgALT  7943  cofunexg  7944  fnexALT  7946  funexw  7947  erssxp  8716  ttrclexg  9690  wunco  10724  trclublem  15039  trclubi  15040  trclub  15042  reltrclfv  15061  imasless  17600  sylow2a  19695  gsum2d  20048  znleval  21715  tsmsxp  24323  relfi  32958  fcnvgreu  33028  elrgspnsubrunlem2  33577  relssinxpdmrn  39026  trclubNEW  44373  trrelsuperreldg  44422  trrelsuperrel2dg  44425  relwf  45704