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

Theorem riotaclb 6689
Description: Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)
Assertion
Ref Expression
riotaclb (¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaclb
StepHypRef Expression
1 riotacl 6665 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
2 riotaund 6687 . . . . . 6 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
32eleq1d 2715 . . . . 5 (¬ ∃!𝑥𝐴 𝜑 → ((𝑥𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴))
43notbid 307 . . . 4 (¬ ∃!𝑥𝐴 𝜑 → (¬ (𝑥𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
54biimprcd 240 . . 3 (¬ ∅ ∈ 𝐴 → (¬ ∃!𝑥𝐴 𝜑 → ¬ (𝑥𝐴 𝜑) ∈ 𝐴))
65con4d 114 . 2 (¬ ∅ ∈ 𝐴 → ((𝑥𝐴 𝜑) ∈ 𝐴 → ∃!𝑥𝐴 𝜑))
71, 6impbid2 216 1 (¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 2030  ∃!wreu 2943  c0 3948  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-riota 6651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator