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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 5623 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | |
2 | df-br 5059 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵))) | |
3 | df-br 5059 | . . 3 ⊢ (𝐶𝑅𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝑅) | |
4 | 3 | anbi2i 624 | . 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr3i 303 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∩ cin 3934 〈cop 4566 class class class wbr 5058 × cxp 5547 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 |
This theorem is referenced by: elinxp 5884 ssrnres 6029 iss2 35595 |
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