Theorem relssres 2596

Description: Simplification law for restriction.

Assertion

Ref	Expression
relssres	⊢ ((Rel A ∧ dom A ⊆ B) → (A ↾ B) = A)

Proof of Theorem relssres

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → Rel A)
2		ssel 1502	. . . . . . . 8 ⊢ (dom A ⊆ B → (x ∈ dom A → x ∈ B))
3		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
4	3	opeldm 2534	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
5	2, 4	syl5 22	. . . . . . 7 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → x ∈ B))
6	5	ancld 246	. . . . . 6 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → (⟨x, y⟩ ∈ A ∧ x ∈ B)))
7		visset 1350	. . . . . . 7 ⊢ y ∈ V
8	7	opelres 2579	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ↾ B) ↔ (⟨x, y⟩ ∈ A ∧ x ∈ B))
9	6, 8	syl6ibr 186	. . . . 5 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (A ↾ B)))
10	9	adantl 305	. . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (A ↾ B)))
11	1, 10	relssdv 2482	. . 3 ⊢ ((Rel A ∧ dom A ⊆ B) → A ⊆ (A ↾ B))
12		resss 2587	. . 3 ⊢ (A ↾ B) ⊆ A
13	11, 12	jctil 240	. 2 ⊢ ((Rel A ∧ dom A ⊆ B) → ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B)))
14		eqss 1516	. 2 ⊢ ((A ↾ B) = A ↔ ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B)))
15	13, 14	sylibr 175	1 ⊢ ((Rel A ∧ dom A ⊆ B) → (A ↾ B) = A)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810  dom cdm 2410   ↾ cres 2412  Rel wrel 2415

This theorem is referenced by:  resid 2601  fnresdm 2731  tz7.48-2 2995  zornlem4 3606  fac0 4871

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-dm 2428  df-res 2430

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