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Theorem dfid7 37745
Description: Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
Assertion
Ref Expression
dfid7 I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfid7
StepHypRef Expression
1 dfid4 5024 . 2 I = (𝑥 ∈ V ↦ 𝑥)
2 ancom 466 . . . . . . 7 ((𝑥𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝑦))
3 truan 1500 . . . . . . 7 ((⊤ ∧ 𝑥𝑦) ↔ 𝑥𝑦)
42, 3bitri 264 . . . . . 6 ((𝑥𝑦 ∧ ⊤) ↔ 𝑥𝑦)
54abbii 2738 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
65inteqi 4477 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
7 vex 3201 . . . . 5 𝑥 ∈ V
87intmin2 4502 . . . 4 {𝑦𝑥𝑦} = 𝑥
96, 8eqtri 2643 . . 3 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = 𝑥
109mpteq2i 4739 . 2 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
111, 10eqtr4i 2646 1 I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1482  wtru 1483  {cab 2607  Vcvv 3198  wss 3572   cint 4473  cmpt 4727   I cid 5021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-in 3579  df-ss 3586  df-int 4474  df-opab 4711  df-mpt 4728  df-id 5022
This theorem is referenced by: (None)
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