Theorem relssdmrn 6104

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 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		19.8a 2179	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		19.8a 2179	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
4		opelxp 5565	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴))
5		vex 3414	. . . . . . 7 ⊢ 𝑥 ∈ V
6	5	eldm2 5748	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
7		vex 3414	. . . . . . 7 ⊢ 𝑦 ∈ V
8	7	elrn2 5739	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
9	6, 8	anbi12i 629	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
10	4, 9	bitri 278	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
11	2, 3, 10	sylanbrc 586	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
12	11	a1i 11	. 2 ⊢ (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
13	1, 12	relssdv 5636	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1782   ∈ wcel 2112   ⊆ wss 3861  ⟨cop 4532   × cxp 5527  dom cdm 5529  ran crn 5530  Rel wrel 5534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5038  df-opab 5100  df-xp 5535  df-rel 5536  df-cnv 5537  df-dm 5539  df-rn 5540

This theorem is referenced by:  resssxp  6105  cnvssrndm  6106  cossxp  6107  relrelss  6108  relfld  6110  fssxp  6525  oprabss  7261  cnvexg  7641  resfunexgALT  7660  cofunexg  7661  fnexALT  7663  funexw  7664  erssxp  8329  wunco  10207  trclublem  14416  trclubi  14417  trclub  14419  reltrclfv  14438  imasless  16886  sylow2a  18826  gsum2d  19175  znleval  20337  tsmsxp  22870  relfi  30478  fcnvgreu  30548  trclubNEW  40738  trrelsuperreldg  40788  trrelsuperrel2dg  40791