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Theorem iresn0n0 5904
Description: The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.)
Assertion
Ref Expression
iresn0n0 (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)

Proof of Theorem iresn0n0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opab0 5422 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
2 opabresid 5898 . . 3 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
32eqeq1i 2829 . 2 (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅)
4 nel02 4279 . . . . 5 (𝐴 = ∅ → ¬ 𝑥𝐴)
54intnanrd 493 . . . 4 (𝐴 = ∅ → ¬ (𝑥𝐴𝑦 = 𝑥))
65alrimivv 1930 . . 3 (𝐴 = ∅ → ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
7 ianor 979 . . . . . . 7 (¬ (𝑥𝐴𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
87albii 1821 . . . . . 6 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) ↔ ∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
9 19.32v 1942 . . . . . . 7 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10 id 22 . . . . . . . 8 𝑥𝐴 → ¬ 𝑥𝐴)
11 ax6v 1972 . . . . . . . . 9 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
1211pm2.21i 119 . . . . . . . 8 (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥𝐴)
1310, 12jaoi 854 . . . . . . 7 ((¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
149, 13sylbi 220 . . . . . 6 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
158, 14sylbi 220 . . . . 5 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ¬ 𝑥𝐴)
1615alimi 1813 . . . 4 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥𝐴)
17 eq0 4289 . . . 4 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
1816, 17sylibr 237 . . 3 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → 𝐴 = ∅)
196, 18impbii 212 . 2 (𝐴 = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
201, 3, 193bitr4ri 307 1 (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844  wal 1536   = wceq 1538  wcel 2115  c0 4274  {copab 5109   I cid 5440  cres 5538
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-res 5548
This theorem is referenced by: (None)
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