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Mirrors > Home > MPE Home > Th. List > relssres | Structured version Visualization version GIF version |
Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
relssres | ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴) | |
2 | vex 3479 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
3 | vex 3479 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opeldm 5908 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
5 | ssel 3976 | . . . . . . . 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 553 | . . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))) |
8 | 3 | opelresi 5990 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
9 | 7, 8 | imbitrrdi 251 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵))) |
10 | 9 | adantl 483 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ 𝐵))) |
11 | 1, 10 | relssdv 5789 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵)) |
12 | resss 6007 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
13 | 11, 12 | jctil 521 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) |
14 | eqss 3998 | . 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) | |
15 | 13, 14 | sylibr 233 | 1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 ⟨cop 4635 dom cdm 5677 ↾ cres 5679 Rel wrel 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-dm 5687 df-res 5689 |
This theorem is referenced by: resdm 6027 fnresdm 6670 focofo 6819 f1ompt 7111 tfr2b 8396 tz7.48-2 8442 omxpenlem 9073 pwfir 9176 rankwflemb 9788 zorn2lem4 10494 relexpaddg 15000 setscom 17113 setsid 17141 dprd2da 19912 dprd2db 19913 ustssco 23719 dvres3 25430 dvres3a 25431 rlimcnp2 26471 nolt02o 27198 nogt01o 27199 nosupbnd1 27217 noinfbnd1 27232 ex-res 29694 symgcom2 32245 poimirlem3 36491 relexpaddss 42469 fnresdmss 43864 limsupresuz 44419 liminfresuz 44500 |
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