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Theorem iresn0n0 6071
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 5558 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
2 opabresid 6067 . . 3 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
32eqeq1i 2741 . 2 (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅)
4 nel02 4338 . . . . 5 (𝐴 = ∅ → ¬ 𝑥𝐴)
54intnanrd 489 . . . 4 (𝐴 = ∅ → ¬ (𝑥𝐴𝑦 = 𝑥))
65alrimivv 1927 . . 3 (𝐴 = ∅ → ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
7 ianor 983 . . . . . . 7 (¬ (𝑥𝐴𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
87albii 1818 . . . . . 6 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) ↔ ∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
9 19.32v 1939 . . . . . . 7 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10 id 22 . . . . . . . 8 𝑥𝐴 → ¬ 𝑥𝐴)
11 ax6v 1967 . . . . . . . . 9 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
1211pm2.21i 119 . . . . . . . 8 (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥𝐴)
1310, 12jaoi 857 . . . . . . 7 ((¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
149, 13sylbi 217 . . . . . 6 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
158, 14sylbi 217 . . . . 5 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ¬ 𝑥𝐴)
1615alimi 1810 . . . 4 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥𝐴)
17 eq0 4349 . . . 4 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
1816, 17sylibr 234 . . 3 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → 𝐴 = ∅)
196, 18impbii 209 . 2 (𝐴 = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
201, 3, 193bitr4ri 304 1 (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  wal 1537   = wceq 1539  wcel 2107  c0 4332  {copab 5204   I cid 5576  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-res 5696
This theorem is referenced by: (None)
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