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Theorem dfid7 40309
 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 5429 . 2 I = (𝑥 ∈ V ↦ 𝑥)
2 ancom 464 . . . . . . 7 ((𝑥𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝑦))
3 truan 1549 . . . . . . 7 ((⊤ ∧ 𝑥𝑦) ↔ 𝑥𝑦)
42, 3bitri 278 . . . . . 6 ((𝑥𝑦 ∧ ⊤) ↔ 𝑥𝑦)
54abbii 2866 . . . . 5 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
65inteqi 4845 . . . 4 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = {𝑦𝑥𝑦}
7 vex 3447 . . . . 5 𝑥 ∈ V
87intmin2 4868 . . . 4 {𝑦𝑥𝑦} = 𝑥
96, 8eqtri 2824 . . 3 {𝑦 ∣ (𝑥𝑦 ∧ ⊤)} = 𝑥
109mpteq2i 5125 . 2 (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
111, 10eqtr4i 2827 1 I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ⊤wtru 1539  {cab 2779  Vcvv 3444   ⊆ wss 3884  ∩ cint 4841   ↦ cmpt 5113   I cid 5427 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-int 4842  df-opab 5096  df-mpt 5114  df-id 5428 This theorem is referenced by: (None)
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