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Theorem bj-imdiridlem 37186
Description: Lemma for bj-imdirid 37187 and bj-iminvid 37196. (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 1471 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝜑) → 𝑥 = 𝑦)
323expib 1123 . . . . . 6 (𝑥𝐴 → ((𝑦𝐴𝜑) → 𝑥 = 𝑦))
4 equcomi 2016 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
54sseq1d 4015 . . . . . . . . 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 4605 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
17 vex 3484 . . . . . 6 𝑦 ∈ V
1817ideq 5863 . . . . 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 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
22 dfres2 6059 . 2 ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
2321, 22eqtr4i 2768 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   class class class wbr 5143  {copab 5205   I cid 5577  cres 5687
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  bj-imdirid  37187  bj-iminvid  37196
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