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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 5381 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | |
2 | df-br 4842 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵))) | |
3 | df-br 4842 | . . 3 ⊢ (𝐶𝑅𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝑅) | |
4 | 3 | anbi2i 617 | . 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr3i 293 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ∩ cin 3766 〈cop 4372 class class class wbr 4841 × cxp 5308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-xp 5316 |
This theorem is referenced by: elinxp 5642 ssrnres 5787 iss2 34597 |
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