Theorem relssdr 2668

Description: A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235.

Assertion

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

Proof of Theorem relssdr

Step	Hyp	Ref	Expression
1		id 9	. 2 ⊢ (Rel A → Rel A)
2		19.8a 712	. . . . 5 ⊢ (⟨x, y⟩ ∈ A → ∃y⟨x, y⟩ ∈ A)
3		19.8a 712	. . . . 5 ⊢ (⟨x, y⟩ ∈ A → ∃x⟨x, y⟩ ∈ A)
4	2, 3	jca 236	. . . 4 ⊢ (⟨x, y⟩ ∈ A → (∃y⟨x, y⟩ ∈ A ∧ ∃x⟨x, y⟩ ∈ A))
5		visset 1350	. . . . . 6 ⊢ y ∈ V
6	5	opelxp 2452	. . . . 5 ⊢ (⟨x, y⟩ ∈ (dom A × ran A) ↔ (x ∈ dom A ∧ y ∈ ran A))
7		visset 1350	. . . . . . 7 ⊢ x ∈ V
8	7	eldm2 2528	. . . . . 6 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
9	5	elrn 2562	. . . . . 6 ⊢ (y ∈ ran A ↔ ∃x⟨x, y⟩ ∈ A)
10	8, 9	anbi12i 369	. . . . 5 ⊢ ((x ∈ dom A ∧ y ∈ ran A) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃x⟨x, y⟩ ∈ A))
11	6, 10	bitr 151	. . . 4 ⊢ (⟨x, y⟩ ∈ (dom A × ran A) ↔ (∃y⟨x, y⟩ ∈ A ∧ ∃x⟨x, y⟩ ∈ A))
12	4, 11	sylibr 175	. . 3 ⊢ (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (dom A × ran A))
13	12	a1i 7	. 2 ⊢ (Rel A → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (dom A × ran A)))
14	1, 13	relssdv 2482	1 ⊢ (Rel A → A ⊆ (dom A × ran A))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   × cxp 2408  dom cdm 2410  ran crn 2411  Rel wrel 2415

This theorem is referenced by:  cnvexg 2669  coexg 2671  resfunexg 2717  fnex 2740  fssxp 2761

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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