Theorem relssdmrn 6289

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 3481	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3481	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5920	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5956	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5727	. . 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 5800	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2105   ⊆ wss 3962  ⟨cop 4636   × cxp 5686  dom cdm 5688  ran crn 5689  Rel wrel 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699

This theorem is referenced by:  resssxp  6291  cnvssrndm  6292  cossxp  6293  relrelss  6294  relfld  6296  fssxp  6763  oprabss  7539  cnvexg  7946  resfunexgALT  7970  cofunexg  7971  fnexALT  7973  funexw  7974  erssxp  8766  ttrclexg  9760  wunco  10770  trclublem  15030  trclubi  15031  trclub  15033  reltrclfv  15052  imasless  17586  sylow2a  19651  gsum2d  20004  znleval  21590  tsmsxp  24178  relfi  32621  fcnvgreu  32689  relssinxpdmrn  38330  trclubNEW  43608  trrelsuperreldg  43657  trrelsuperrel2dg  43660  relwf  44941