MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iresn0n0 Structured version   Visualization version   GIF version

Theorem iresn0n0 6074
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 5564 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
2 opabresid 6070 . . 3 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
32eqeq1i 2740 . 2 (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅)
4 nel02 4345 . . . . 5 (𝐴 = ∅ → ¬ 𝑥𝐴)
54intnanrd 489 . . . 4 (𝐴 = ∅ → ¬ (𝑥𝐴𝑦 = 𝑥))
65alrimivv 1926 . . 3 (𝐴 = ∅ → ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
7 ianor 983 . . . . . . 7 (¬ (𝑥𝐴𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
87albii 1816 . . . . . 6 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) ↔ ∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
9 19.32v 1938 . . . . . . 7 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10 id 22 . . . . . . . 8 𝑥𝐴 → ¬ 𝑥𝐴)
11 ax6v 1966 . . . . . . . . 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 1808 . . . 4 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥𝐴)
17 eq0 4356 . . . 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 1535   = wceq 1537  wcel 2106  c0 4339  {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-10 2139  ax-11 2155  ax-12 2175  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-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  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-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator