Theorem bj-imdiridlem 37390

Description: Lemma for bj-imdirid 37391 and bj-iminvid 37400. (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 1471	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)
3	2	3expib 1122	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
4		equcomi 2018	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
5	4	sseq1d 3965	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑦 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
6	5	biimparc 479	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) → 𝑦 ⊆ 𝐴)
7		simpr 484	. . . . . . . . 9 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
8	1	biimpar 477	. . . . . . . . . 10 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = 𝑦) → 𝜑)
9	8	an32s 652	. . . . . . . . 9 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝜑)
10	7, 9	jca 511	. . . . . . . 8 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ∧ 𝑦 ⊆ 𝐴) → (𝑦 ⊆ 𝐴 ∧ 𝜑))
11	6, 10	mpdan 687	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦) → (𝑦 ⊆ 𝐴 ∧ 𝜑))
12	11	ex 412	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 = 𝑦 → (𝑦 ⊆ 𝐴 ∧ 𝜑)))
13	3, 12	impbid 212	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑦))
14	13	pm5.32i 574	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝜑)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
15		anass 468	. . . 4 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑) ↔ (𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝜑)))
16		velpw 4559	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
17		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
18	17	ideq 5801	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
19	16, 18	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
20	14, 15, 19	3bitr4i 303	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦))
21	20	opabbii 5165	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
22		dfres2 6000	. 2 ⊢ ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
23	21, 22	eqtr4i 2762	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098  {copab 5160   I cid 5518   ↾ cres 5626

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-res 5636

This theorem is referenced by:  bj-imdirid  37391  bj-iminvid  37400