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Mirrors > Home > MPE Home > Th. List > opelinxp | Structured version Visualization version GIF version |
Description: Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.) |
Ref | Expression |
---|---|
opelinxp | ⊢ (〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brinxp2 5766 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | |
2 | df-br 5149 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵))) | |
3 | df-br 5149 | . . 3 ⊢ (𝐶𝑅𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝑅) | |
4 | 3 | anbi2i 623 | . 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr3i 301 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∩ cin 3962 〈cop 4637 class class class wbr 5148 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 |
This theorem is referenced by: elinxp 6039 ssrnres 6200 iss2 38326 |
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