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

Theorem iresn0n0 6011
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 5515 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
2 opabresid 6007 . . 3 ( I ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)}
32eqeq1i 2738 . 2 (( I ↾ 𝐴) = ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ∅)
4 nel02 4296 . . . . 5 (𝐴 = ∅ → ¬ 𝑥𝐴)
54intnanrd 491 . . . 4 (𝐴 = ∅ → ¬ (𝑥𝐴𝑦 = 𝑥))
65alrimivv 1932 . . 3 (𝐴 = ∅ → ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
7 ianor 981 . . . . . . 7 (¬ (𝑥𝐴𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
87albii 1822 . . . . . 6 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) ↔ ∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥))
9 19.32v 1944 . . . . . . 7 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) ↔ (¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥))
10 id 22 . . . . . . . 8 𝑥𝐴 → ¬ 𝑥𝐴)
11 ax6v 1973 . . . . . . . . 9 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
1211pm2.21i 119 . . . . . . . 8 (∀𝑦 ¬ 𝑦 = 𝑥 → ¬ 𝑥𝐴)
1310, 12jaoi 856 . . . . . . 7 ((¬ 𝑥𝐴 ∨ ∀𝑦 ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
149, 13sylbi 216 . . . . . 6 (∀𝑦𝑥𝐴 ∨ ¬ 𝑦 = 𝑥) → ¬ 𝑥𝐴)
158, 14sylbi 216 . . . . 5 (∀𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ¬ 𝑥𝐴)
1615alimi 1814 . . . 4 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → ∀𝑥 ¬ 𝑥𝐴)
17 eq0 4307 . . . 4 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
1816, 17sylibr 233 . . 3 (∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥) → 𝐴 = ∅)
196, 18impbii 208 . 2 (𝐴 = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝐴𝑦 = 𝑥))
201, 3, 193bitr4ri 304 1 (𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846  wal 1540   = wceq 1542  wcel 2107  c0 4286  {copab 5171   I cid 5534  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-res 5649
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator