Theorem relssdmrn 6244

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 22	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5874	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5910	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5680	. . 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 5754	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3917  ⟨cop 4598   × cxp 5639  dom cdm 5641  ran crn 5642  Rel wrel 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652

This theorem is referenced by:  resssxp  6246  cnvssrndm  6247  cossxp  6248  relrelss  6249  relfld  6251  fssxp  6718  oprabss  7500  cnvexg  7903  resfunexgALT  7929  cofunexg  7930  fnexALT  7932  funexw  7933  erssxp  8697  ttrclexg  9683  wunco  10693  trclublem  14968  trclubi  14969  trclub  14971  reltrclfv  14990  imasless  17510  sylow2a  19556  gsum2d  19909  znleval  21471  tsmsxp  24049  relfi  32538  fcnvgreu  32604  elrgspnsubrunlem2  33206  relssinxpdmrn  38338  trclubNEW  43615  trrelsuperreldg  43664  trrelsuperrel2dg  43667  relwf  44964