Theorem dfid7 44264

Description: Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)

Assertion

Ref	Expression
dfid7	⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfid7

Step	Hyp	Ref	Expression
1		dfid4 5558	. 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
2		ancom 465	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦))
3		truan 1578	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦)
4	2, 3	bitri 278	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦)
5	4	abbii 2836	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦}
6	5	inteqi 4920	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦}
7		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
8	7	intmin2 4944	. . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥
9	6, 8	eqtri 2792	. . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥
10	9	mpteq2i 5211	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
11	1, 10	eqtr4i 2795	1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   ∧ wa 400   = wceq 1567  ⊤wtru 1568  {cab 2747  Vcvv 3463   ⊆ wss 3913  ∩ cint 4916   ↦ cmpt 5196   I cid 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-ss 3930  df-int 4917  df-opab 5178  df-mpt 5197  df-id 5557

This theorem is referenced by: (None)