Theorem riotaclbBAD 39584

Description: Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)

Hypothesis

Ref	Expression
riotaclb.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
riotaclbBAD	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaclbBAD

Step	Hyp	Ref	Expression
1		riotaclb.1	. 2 ⊢ 𝐴 ∈ V
2		riotaclbgBAD 39583	. 2 ⊢ (𝐴 ∈ V → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2144  ∃!wreu 3367  Vcvv 3456  ℩crio 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-riotaBAD 39582

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-riota 7355  df-undef 8255

This theorem is referenced by:  cdlemk36  41542