Theorem relssres 6042

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 482	. . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴)
2		vex 3482	. . . . . . . . 9 ⊢ 𝑥 ∈ V
3		vex 3482	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5921	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5		ssel 3989	. . . . . . . 8 ⊢ (dom 𝐴 ⊆ 𝐵 → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ 𝐵))
6	4, 5	syl5 34	. . . . . . 7 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	6	ancrd 551	. . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
8	3	opelresi 6008	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
9	7, 8	imbitrrdi 252	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵)))
10	9	adantl 481	. . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵)))
11	1, 10	relssdv 5801	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵))
12		resss 6022	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
13	11, 12	jctil 519	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵)))
14		eqss 4011	. 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵)))
15	13, 14	sylibr 234	1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ⟨cop 4637  dom cdm 5689   ↾ cres 5691  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701

This theorem is referenced by:  resdm  6046  fnresdm  6688  focofo  6834  f1ompt  7131  tfr2b  8435  tz7.48-2  8481  omxpenlem  9112  pwfir  9353  rankwflemb  9831  zorn2lem4  10537  relexpaddg  15089  setscom  17214  setsid  17242  dprd2da  20077  dprd2db  20078  ustssco  24239  dvres3  25963  dvres3a  25964  rlimcnp2  27024  nolt02o  27755  nogt01o  27756  nosupbnd1  27774  noinfbnd1  27789  ex-res  30470  symgcom2  33087  poimirlem3  37610  relexpaddss  43708  fnresdmss  45111  limsupresuz  45659  liminfresuz  45740  isubgrvtxuhgr  47788