Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfid7 Structured version   Visualization version   GIF version

Theorem dfid7 38439
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 5176 . 2 I = (𝑥 ∈ V ↦ 𝑥)
2 ancom 465 . . . . . . 7 ((𝑥𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝑦))
3 truan 1650 . . . . . . 7 ((⊤ ∧ 𝑥𝑦) ↔ 𝑥𝑦)
42, 3bitri 264 . . . . . 6 ((𝑥𝑦 ∧ ⊤) ↔ 𝑥𝑦)
54abbii 2877 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
65inteqi 4631 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
7 vex 3343 . . . . 5 𝑥 ∈ V
87intmin2 4656 . . . 4 {𝑦𝑥𝑦} = 𝑥
96, 8eqtri 2782 . . 3 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = 𝑥
109mpteq2i 4893 . 2 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
111, 10eqtr4i 2785 1 I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wtru 1633  {cab 2746  Vcvv 3340  wss 3715   cint 4627  cmpt 4881   I cid 5173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-in 3722  df-ss 3729  df-int 4628  df-opab 4865  df-mpt 4882  df-id 5174
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator