Theorem dfid7 44366

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 5557	. 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥)
2		ancom 465	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦))
3		truan 1581	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦)
4	2, 3	bitri 278	. . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦)
5	4	abbii 2830	. . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦}
6	5	inteqi 4916	. . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦}
7		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
8	7	intmin2 4940	. . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥
9	6, 8	eqtri 2786	. . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥
10	9	mpteq2i 5207	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥)
11	1, 10	eqtr4i 2789	1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1570  ⊤wtru 1571  {cab 2741  Vcvv 3455   ⊆ wss 3905  ∩ cint 4912   ↦ cmpt 5192   I cid 5555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-ss 3922  df-int 4913  df-opab 5174  df-mpt 5193  df-id 5556

This theorem is used by: (None)