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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 | brinxp2ALTV 34377 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | |
2 | df-br 4787 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵))) | |
3 | df-br 4787 | . . 3 ⊢ (𝐶𝑅𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝑅) | |
4 | 3 | anbi2i 609 | . 2 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr3i 290 | 1 ⊢ (〈𝐶, 𝐷〉 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 〈𝐶, 𝐷〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∈ wcel 2145 ∩ cin 3722 〈cop 4322 class class class wbr 4786 × cxp 5247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 |
This theorem is referenced by: iss2 34454 |
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