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Theorem dfid7 41683
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 5530 . 2 I = (𝑥 ∈ V ↦ 𝑥)
2 ancom 462 . . . . . . 7 ((𝑥𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝑦))
3 truan 1553 . . . . . . 7 ((⊤ ∧ 𝑥𝑦) ↔ 𝑥𝑦)
42, 3bitri 275 . . . . . 6 ((𝑥𝑦 ∧ ⊤) ↔ 𝑥𝑦)
54abbii 2808 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
65inteqi 4910 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
7 vex 3448 . . . . 5 𝑥 ∈ V
87intmin2 4935 . . . 4 {𝑦𝑥𝑦} = 𝑥
96, 8eqtri 2766 . . 3 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = 𝑥
109mpteq2i 5209 . 2 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
111, 10eqtr4i 2769 1 I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wtru 1543  {cab 2715  Vcvv 3444  wss 3909   cint 4906  cmpt 5187   I cid 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rab 3407  df-v 3446  df-in 3916  df-ss 3926  df-int 4907  df-opab 5167  df-mpt 5188  df-id 5529
This theorem is referenced by: (None)
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