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Theorem elidinxpid 5884
 Description: Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elidinxpid (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elidinxpid
StepHypRef Expression
1 elidinxp 5883 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2 inidm 4123 . . 3 (𝐴𝐴) = 𝐴
32rexeqi 3328 . 2 (∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
41, 3bitri 278 1 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3857  ⟨cop 4528   I cid 5429   × cxp 5522 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531 This theorem is referenced by: (None)
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