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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid7 | Structured version Visualization version GIF version |
Description: Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-elid7 | ⊢ (〈𝐵, 𝐶〉 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5074 | . 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ 〈𝐵, 𝐶〉 ∈ ( I ↾ 𝐴)) | |
2 | bj-idreseqb 35342 | . 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) | |
3 | 1, 2 | bitr3i 276 | 1 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 〈cop 4567 class class class wbr 5073 I cid 5483 ↾ cres 5586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 df-res 5596 |
This theorem is referenced by: (None) |
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