Theorem relssres 4591

Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Assertion

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

Proof of Theorem relssres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 102	. . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → Rel A)
2		vex 2554	. . . . . . . . 9 ⊢ x ∈ V
3		vex 2554	. . . . . . . . 9 ⊢ y ∈ V
4	2, 3	opeldm 4481	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
5		ssel 2933	. . . . . . . 8 ⊢ (dom A ⊆ B → (x ∈ dom A → x ∈ B))
6	4, 5	syl5 28	. . . . . . 7 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → x ∈ B))
7	6	ancld 308	. . . . . 6 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → (⟨x, y⟩ ∈ A ∧ x ∈ B)))
8	3	opelres 4560	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ↾ B) ↔ (⟨x, y⟩ ∈ A ∧ x ∈ B))
9	7, 8	syl6ibr 151	. . . . 5 ⊢ (dom A ⊆ B → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (A ↾ B)))
10	9	adantl 262	. . . 4 ⊢ ((Rel A ∧ dom A ⊆ B) → (⟨x, y⟩ ∈ A → ⟨x, y⟩ ∈ (A ↾ B)))
11	1, 10	relssdv 4375	. . 3 ⊢ ((Rel A ∧ dom A ⊆ B) → A ⊆ (A ↾ B))
12		resss 4578	. . 3 ⊢ (A ↾ B) ⊆ A
13	11, 12	jctil 295	. 2 ⊢ ((Rel A ∧ dom A ⊆ B) → ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B)))
14		eqss 2954	. 2 ⊢ ((A ↾ B) = A ↔ ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B)))
15	13, 14	sylibr 137	1 ⊢ ((Rel A ∧ dom A ⊆ B) → (A ↾ B) = A)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ⟨cop 3370  dom cdm 4288   ↾ cres 4290  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300

This theorem is referenced by:  resdm  4592  resid  4605  fnresdm  4951  f1ompt  5263