Theorem dfid7 43715

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 5510	. 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
2		ancom 460	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦))
3		truan 1552	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦)
4	2, 3	bitri 275	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦)
5	4	abbii 2798	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦}
6	5	inteqi 4899	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦}
7		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
8	7	intmin2 4923	. . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥
9	6, 8	eqtri 2754	. . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥
10	9	mpteq2i 5185	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
11	1, 10	eqtr4i 2757	1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   ∧ wa 395   = wceq 1541  ⊤wtru 1542  {cab 2709  Vcvv 3436   ⊆ wss 3897  ∩ cint 4895   ↦ cmpt 5170   I cid 5508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-ss 3914  df-int 4896  df-opab 5152  df-mpt 5171  df-id 5509

This theorem is referenced by: (None)