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Theorem idinxpres 5697
 Description: The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
idinxpres ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)

Proof of Theorem idinxpres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elidinxpid 5694 . . 3 (𝑦 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
2 elrid 5695 . . 3 (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
31, 2bitr4i 270 . 2 (𝑦 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ 𝑦 ∈ ( I ↾ 𝐴))
43eqriv 2823 1 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ∈ wcel 2166  ∃wrex 3119   ∩ cin 3798  ⟨cop 4404   I cid 5250   × cxp 5341   ↾ cres 5345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-res 5355 This theorem is referenced by:  idssxp  5698  idinxpssinxp2  34638
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