Theorem dfid7 42134

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 5568	. 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
2		ancom 461	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦))
3		truan 1552	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦)
4	2, 3	bitri 274	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦)
5	4	abbii 2801	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦}
6	5	inteqi 4947	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦}
7		vex 3477	. . . . 5 ⊢ 𝑥 ∈ V
8	7	intmin2 4972	. . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥
9	6, 8	eqtri 2759	. . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥
10	9	mpteq2i 5246	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
11	1, 10	eqtr4i 2762	1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   ∧ wa 396   = wceq 1541  ⊤wtru 1542  {cab 2708  Vcvv 3473   ⊆ wss 3944  ∩ cint 4943   ↦ cmpt 5224   I cid 5566

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-in 3951  df-ss 3961  df-int 4944  df-opab 5204  df-mpt 5225  df-id 5567

This theorem is referenced by: (None)