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Theorem elidinxpid 5693
 Description: Characterization of elements of the intersection of identity relation with square Cartesian product. (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 5692 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2 inidm 4047 . . 3 (𝐴𝐴) = 𝐴
32rexeqi 3355 . 2 (∃𝑥 ∈ (𝐴𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
41, 3bitri 267 1 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1656   ∈ wcel 2164  ∃wrex 3118   ∩ cin 3797  ⟨cop 4403   I cid 5249   × cxp 5340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349 This theorem is referenced by:  idinxpres  5696
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