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| Mirrors > Home > MPE Home > Th. List > opelres | Structured version Visualization version GIF version | ||
| Description: Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| opelres | ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-res 5696 | . . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
| 2 | 1 | elin2 4202 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (〈𝐵, 𝐶〉 ∈ 𝑅 ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × V))) | 
| 3 | opelxp 5720 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × V) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V)) | |
| 4 | elex 3500 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
| 5 | 4 | biantrud 531 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝐵 ∈ 𝐴 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V))) | 
| 6 | 3, 5 | bitr4id 290 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝐴 × V) ↔ 𝐵 ∈ 𝐴)) | 
| 7 | 6 | anbi1cd 635 | . 2 ⊢ (𝐶 ∈ 𝑉 → ((〈𝐵, 𝐶〉 ∈ 𝑅 ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × V)) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | 
| 8 | 2, 7 | bitrid 283 | 1 ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 〈cop 4631 × cxp 5682 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-res 5696 | 
| This theorem is referenced by: brres 6003 opelresi 6004 opelidres 6008 h2hlm 31000 setsnidel 47369 | 
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