Theorem relssres 4571

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 2534	. . . . . . . . 9 ⊢ x ∈ V
3		vex 2534	. . . . . . . . 9 ⊢ y ∈ V
4	2, 3	opeldm 4461	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ A → x ∈ dom A)
5		ssel 2912	. . . . . . . 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 4540	. . . . . 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 4355	. . 3 ⊢ ((Rel A ∧ dom A ⊆ B) → A ⊆ (A ↾ B))
12		resss 4558	. . 3 ⊢ (A ↾ B) ⊆ A
13	11, 12	jctil 295	. 2 ⊢ ((Rel A ∧ dom A ⊆ B) → ((A ↾ B) ⊆ A ∧ A ⊆ (A ↾ B)))
14		eqss 2933	. 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 1226   ∈ wcel 1370   ⊆ wss 2890  ⟨cop 3349  dom cdm 4268   ↾ cres 4270  Rel wrel 4273

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-dm 4278  df-res 4280

This theorem is referenced by:  resdm  4572  resid  4585  fnresdm  4930  f1ompt  5241