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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 5679 | . . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V)) | |
2 | 1 | elin2 4190 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (〈𝐵, 𝐶〉 ∈ 𝑅 ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × V))) |
3 | opelxp 5703 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × V) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V)) | |
4 | elex 3485 | . . . . 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 633 | . 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 205 ∧ wa 395 ∈ wcel 2098 Vcvv 3466 〈cop 4627 × cxp 5665 ↾ cres 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-opab 5202 df-xp 5673 df-res 5679 |
This theorem is referenced by: brres 5979 opelresi 5980 opelidres 5984 h2hlm 30705 setsnidel 46555 |
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