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Mirrors > Home > MPE Home > Th. List > elidinxpid | Structured version Visualization version GIF version |
Description: Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.) |
Ref | Expression |
---|---|
elidinxpid | ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = 〈𝑥, 𝑥〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elidinxp 5996 | . 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = 〈𝑥, 𝑥〉) | |
2 | inidm 4177 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
3 | 2 | rexeqi 3311 | . 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 ∈ 𝐴 𝐵 = 〈𝑥, 𝑥〉) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ∩ cin 3908 〈cop 4591 I cid 5529 × cxp 5630 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 |
This theorem is referenced by: (None) |
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