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Theorem intiin 3711
 Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
intiin 𝐴 = 𝑥𝐴 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem intiin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfint2 3617 . 2 𝐴 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
2 df-iin 3660 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2063 1 𝐴 = 𝑥𝐴 𝑥
 Colors of variables: wff set class Syntax hints:   = wceq 1243  {cab 2026  ∀wral 2306  ∩ cint 3615  ∩ ciin 3658 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-ral 2311  df-int 3616  df-iin 3660 This theorem is referenced by:  relint  4461
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