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Theorem bj-imdiridlem 37168
Description: Lemma for bj-imdirid 37169 and bj-iminvid 37178. (Contributed by BJ, 26-May-2024.)
Hypothesis
Ref Expression
bj-imdiridlem.1 ((𝑥𝐴𝑦𝐴) → (𝜑𝑥 = 𝑦))
Assertion
Ref Expression
bj-imdiridlem {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-imdiridlem
StepHypRef Expression
1 bj-imdiridlem.1 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝜑𝑥 = 𝑦))
21biimp3a 1468 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝜑) → 𝑥 = 𝑦)
323expib 1121 . . . . . 6 (𝑥𝐴 → ((𝑦𝐴𝜑) → 𝑥 = 𝑦))
4 equcomi 2014 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
54sseq1d 4027 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑦𝐴𝑥𝐴))
65biimparc 479 . . . . . . . 8 ((𝑥𝐴𝑥 = 𝑦) → 𝑦𝐴)
7 simpr 484 . . . . . . . . 9 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → 𝑦𝐴)
81biimpar 477 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑥 = 𝑦) → 𝜑)
98an32s 652 . . . . . . . . 9 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → 𝜑)
107, 9jca 511 . . . . . . . 8 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → (𝑦𝐴𝜑))
116, 10mpdan 687 . . . . . . 7 ((𝑥𝐴𝑥 = 𝑦) → (𝑦𝐴𝜑))
1211ex 412 . . . . . 6 (𝑥𝐴 → (𝑥 = 𝑦 → (𝑦𝐴𝜑)))
133, 12impbid 212 . . . . 5 (𝑥𝐴 → ((𝑦𝐴𝜑) ↔ 𝑥 = 𝑦))
1413pm5.32i 574 . . . 4 ((𝑥𝐴 ∧ (𝑦𝐴𝜑)) ↔ (𝑥𝐴𝑥 = 𝑦))
15 anass 468 . . . 4 (((𝑥𝐴𝑦𝐴) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝜑)))
16 velpw 4610 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
17 vex 3482 . . . . . 6 𝑦 ∈ V
1817ideq 5866 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1916, 18anbi12i 628 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦) ↔ (𝑥𝐴𝑥 = 𝑦))
2014, 15, 193bitr4i 303 . . 3 (((𝑥𝐴𝑦𝐴) ∧ 𝜑) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦))
2120opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
22 dfres2 6061 . 2 ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
2321, 22eqtr4i 2766 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605   class class class wbr 5148  {copab 5210   I cid 5582  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  bj-imdirid  37169  bj-iminvid  37178
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