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Theorem riotaclb 7410
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 7386 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ∈ 𝐴)
2 riotaund 7408 . . . . . 6 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
32eleq1d 2817 . . . . 5 (¬ ∃!𝑥𝐴 𝜑 → ((𝑥𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴))
43notbid 318 . . . 4 (¬ ∃!𝑥𝐴 𝜑 → (¬ (𝑥𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
54biimprcd 249 . . 3 (¬ ∅ ∈ 𝐴 → (¬ ∃!𝑥𝐴 𝜑 → ¬ (𝑥𝐴 𝜑) ∈ 𝐴))
65con4d 115 . 2 (¬ ∅ ∈ 𝐴 → ((𝑥𝐴 𝜑) ∈ 𝐴 → ∃!𝑥𝐴 𝜑))
71, 6impbid2 225 1 (¬ ∅ ∈ 𝐴 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2105  ∃!wreu 3373  c0 4322  crio 7367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-riota 7368
This theorem is referenced by: (None)
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