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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 482 | . . . 4 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → Rel 𝐴) | |
| 2 | vex 3484 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | opeldm 5918 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
| 5 | ssel 3977 | . . . . . . . 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 6005 | . . . . . 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 5798 | . . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐴 ↾ 𝐵)) |
| 12 | resss 6019 | . . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
| 13 | 11, 12 | jctil 519 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) |
| 14 | eqss 3999 | . 2 ⊢ ((𝐴 ↾ 𝐵) = 𝐴 ↔ ((𝐴 ↾ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ↾ 𝐵))) | |
| 15 | 13, 14 | sylibr 234 | 1 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) → (𝐴 ↾ 𝐵) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 |
| This theorem is referenced by: resdm 6044 fnresdm 6687 focofo 6833 f1ompt 7131 tfr2b 8436 tz7.48-2 8482 omxpenlem 9113 pwfir 9355 rankwflemb 9833 zorn2lem4 10539 relexpaddg 15092 setscom 17217 setsid 17244 dprd2da 20062 dprd2db 20063 ustssco 24223 dvres3 25948 dvres3a 25949 rlimcnp2 27009 nolt02o 27740 nogt01o 27741 nosupbnd1 27759 noinfbnd1 27774 ex-res 30460 symgcom2 33104 poimirlem3 37630 relexpaddss 43731 fnresdmss 45173 limsupresuz 45718 liminfresuz 45799 isubgrvtxuhgr 47850 tposresg 48778 |
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