Theorem relssdmrn 5264

Description: A relation is included in the cross 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 19	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		19.8a 1639	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		19.8a 1639	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
4		opelxp 4761	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴))
5		vex 2806	. . . . . . 7 ⊢ 𝑥 ∈ V
6	5	eldm2 4935	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
7		vex 2806	. . . . . . 7 ⊢ 𝑦 ∈ V
8	7	elrn2 4980	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
9	6, 8	anbi12i 460	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
10	4, 9	bitri 184	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
11	2, 3, 10	sylanbrc 417	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
12	11	a1i 9	. 2 ⊢ (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
13	1, 12	relssdv 4824	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1541   ∈ wcel 2202   ⊆ wss 3201  ⟨cop 3676   × cxp 4729  dom cdm 4731  ran crn 4732  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  cnvssrndm  5265  cossxp  5266  relrelss  5270  relfld  5272  cnvexg  5281  fssxp  5510  oprabss  6117  resfunexgALT  6279  cofunexg  6280  fnexALT  6282  funexw  6283  erssxp  6768  znleval  14749