Theorem bj-imdiridlem 37173

Description: Lemma for bj-imdirid 37174 and bj-iminvid 37183. (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 2017	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
5	4	sseq1d 3978	. . . . . . . . 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 4568	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
17		vex 3451	. . . . . 6 ⊢ 𝑦 ∈ V
18	17	ideq 5816	. . . . 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 5174	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
22		dfres2 6012	. 2 ⊢ ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
23	21, 22	eqtr4i 2755	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  𝒫 cpw 4563   class class class wbr 5107  {copab 5169   I cid 5532   ↾ cres 5640

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-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-res 5650

This theorem is referenced by:  bj-imdirid  37174  bj-iminvid  37183