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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcnvin | Structured version Visualization version GIF version |
Description: Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.) |
Ref | Expression |
---|---|
brcnvin | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brin 5157 | . 2 ⊢ (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵)) | |
2 | brcnvg 5835 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡𝑆𝐵 ↔ 𝐵𝑆𝐴)) | |
3 | 2 | anbi2d 629 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴))) |
4 | 1, 3 | bitrid 282 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∩ cin 3909 class class class wbr 5105 ◡ccnv 5632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-cnv 5641 |
This theorem is referenced by: dfantisymrel5 37224 |
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