Theorem relssres 4947

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 109	. . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴)
2		vex 2742	. . . . . . . . 9 ⊢ 𝑥 ∈ V
3		vex 2742	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 4832	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5		ssel 3151	. . . . . . . 8 ⊢ (dom 𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ 𝐵))
6	4, 5	syl5 32	. . . . . . 7 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	6	ancld 325	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
8	3	opelres 4914	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
9	7, 8	imbitrrdi 162	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵)))
10	9	adantl 277	. . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵)))
11	1, 10	relssdv 4720	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵))
12		resss 4933	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
13	11, 12	jctil 312	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵)))
14		eqss 3172	. 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵)))
15	13, 14	sylibr 134	1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  ⟨cop 3597  dom cdm 4628   ↾ cres 4630  Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-dm 4638  df-res 4640

This theorem is referenced by:  resdm  4948  resid  4966  fnresdm  5327  f1ompt  5669  setscom  12504  setsslid  12515