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Theorem riotaclbBAD 36200
Description: Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaclb.1 𝐴 ∈ V
Assertion
Ref Expression
riotaclbBAD (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaclbBAD
StepHypRef Expression
1 riotaclb.1 . 2 𝐴 ∈ V
2 riotaclbgBAD 36199 . 2 (𝐴 ∈ V → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
31, 2ax-mp 5 1 (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  ∃!wreu 3135  Vcvv 3480  crio 7106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-riotaBAD 36198
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-riota 7107  df-undef 7935
This theorem is referenced by:  glbconN  36622  cdlemk36  38158
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