Theorem bj-imdiridlem 37545

Description: Lemma for bj-imdirid 37546 and bj-iminvid 37555. (Contributed by BJ, 26-May-2024.)

Hypothesis

Ref	Expression
bj-imdiridlem.1	⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝜑 ↔ 𝑥 = 𝑦))

Assertion

Ref	Expression
bj-imdiridlem	⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-imdiridlem

Step	Hyp	Ref	Expression
1		bj-imdiridlem.1	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝜑 ↔ 𝑥 = 𝑦))
2	1	biimp3a 1477	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)
3	2	3expib 1128	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
4		equcomi 2024	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
5	4	sseq1d 3946	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑦 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
6	5	biimparc 480	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) → 𝑦 ⊆ 𝐴)
7		simpr 485	. . . . . . . . 9 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
8	1	biimpar 478	. . . . . . . . . 10 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = 𝑦) → 𝜑)
9	8	an32s 658	. . . . . . . . 9 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝜑)
10	7, 9	jca 516	. . . . . . . 8 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → (𝑦 ⊆ 𝐴 ∧ 𝜑))
11	6, 10	mpdan 693	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) → (𝑦 ⊆ 𝐴 ∧ 𝜑))
12	11	ex 413	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 = 𝑦 → (𝑦 ⊆ 𝐴 ∧ 𝜑)))
13	3, 12	impbid 213	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
14	13	pm5.32i 579	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝜑)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
15		anass 469	. . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑) ↔ (𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝜑)))
16		velpw 4534	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
17		vex 3435	. . . . . 6 ⊢ 𝑦 ∈ V
18	17	ideq 5794	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	16, 18	anbi12i 634	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
20	14, 15, 19	3bitr4i 304	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦))
21	20	opabbii 5139	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
22		dfres2 5993	. 2 ⊢ ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
23	21, 22	eqtr4i 2765	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  𝒫 cpw 4529   class class class wbr 5072  {copab 5134   I cid 5512   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-res 5630

This theorem is referenced by:  bj-imdirid  37546  bj-iminvid  37555