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Theorem bj-imdiridlem 34561
Description: Lemma for bj-imdirid 34562 and bj-iminvid 34571. (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 1466 . . . . . . 7 ((𝑥𝐴𝑦𝐴𝜑) → 𝑥 = 𝑦)
323expib 1119 . . . . . 6 (𝑥𝐴 → ((𝑦𝐴𝜑) → 𝑥 = 𝑦))
4 equcomi 2024 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
54sseq1d 3973 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑦𝐴𝑥𝐴))
65biimparc 483 . . . . . . . 8 ((𝑥𝐴𝑥 = 𝑦) → 𝑦𝐴)
7 simpr 488 . . . . . . . . 9 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → 𝑦𝐴)
81biimpar 481 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐴) ∧ 𝑥 = 𝑦) → 𝜑)
98an32s 651 . . . . . . . . 9 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → 𝜑)
107, 9jca 515 . . . . . . . 8 (((𝑥𝐴𝑥 = 𝑦) ∧ 𝑦𝐴) → (𝑦𝐴𝜑))
116, 10mpdan 686 . . . . . . 7 ((𝑥𝐴𝑥 = 𝑦) → (𝑦𝐴𝜑))
1211ex 416 . . . . . 6 (𝑥𝐴 → (𝑥 = 𝑦 → (𝑦𝐴𝜑)))
133, 12impbid 215 . . . . 5 (𝑥𝐴 → ((𝑦𝐴𝜑) ↔ 𝑥 = 𝑦))
1413pm5.32i 578 . . . 4 ((𝑥𝐴 ∧ (𝑦𝐴𝜑)) ↔ (𝑥𝐴𝑥 = 𝑦))
15 anass 472 . . . 4 (((𝑥𝐴𝑦𝐴) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐴𝜑)))
16 velpw 4516 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
17 vex 3472 . . . . . 6 𝑦 ∈ V
1817ideq 5700 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1916, 18anbi12i 629 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦) ↔ (𝑥𝐴𝑥 = 𝑦))
2014, 15, 193bitr4i 306 . . 3 (((𝑥𝐴𝑦𝐴) ∧ 𝜑) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦))
2120opabbii 5109 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
22 dfres2 5887 . 2 ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑥 I 𝑦)}
2321, 22eqtr4i 2848 1 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wss 3908  𝒫 cpw 4511   class class class wbr 5042  {copab 5104   I cid 5436  cres 5534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-res 5544
This theorem is referenced by:  bj-imdirid  34562  bj-iminvid  34571
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