Theorem dfid7 43608

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 5537	. 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
2		ancom 460	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦))
3		truan 1551	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦)
4	2, 3	bitri 275	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦)
5	4	abbii 2797	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦}
6	5	inteqi 4917	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦}
7		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
8	7	intmin2 4942	. . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥
9	6, 8	eqtri 2753	. . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥
10	9	mpteq2i 5206	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
11	1, 10	eqtr4i 2756	1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   ∧ wa 395   = wceq 1540  ⊤wtru 1541  {cab 2708  Vcvv 3450   ⊆ wss 3917  ∩ cint 4913   ↦ cmpt 5191   I cid 5535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-ss 3934  df-int 4914  df-opab 5173  df-mpt 5192  df-id 5536

This theorem is referenced by: (None)