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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 476 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴) | |
2 | vex 3417 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
3 | vex 3417 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opeldm 5564 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
5 | ssel 3821 | . . . . . . . 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 547 | . . . . . 6 ⊢ (dom 𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
8 | 3 | opelresi 5641 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
9 | 7, 8 | syl6ibr 244 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐵 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵))) |
10 | 9 | adantl 475 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (𝐴 ↾ 𝐵))) |
11 | 1, 10 | relssdv 5450 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵)) |
12 | resss 5662 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
13 | 11, 12 | jctil 515 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) |
14 | eqss 3842 | . 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) | |
15 | 13, 14 | sylibr 226 | 1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 〈cop 4405 dom cdm 5346 ↾ cres 5348 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-dm 5356 df-res 5358 |
This theorem is referenced by: resdm 5682 fnresdm 6237 f1ompt 6635 tfr2b 7763 tz7.48-2 7808 omxpenlem 8336 rankwflemb 8940 zorn2lem4 9643 relexpaddg 14177 setscom 16273 setsid 16284 dprd2da 18802 dprd2db 18803 ustssco 22395 dvres3 24083 dvres3a 24084 rlimcnp2 25113 ex-res 27852 nolt02o 32379 nosupbnd1 32394 poimirlem3 33955 relexpaddss 38850 fnresdmss 40156 limsupresuz 40728 liminfresuz 40809 |
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