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| 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 5763 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | |
| 2 | df-br 5144 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵))) | |
| 3 | df-br 5144 | . . 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 2108 ∩ cin 3950 〈cop 4632 class class class wbr 5143 × cxp 5683 | 
| 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 | 
| This theorem is referenced by: elinxp 6037 ssrnres 6198 iss2 38345 | 
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