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Theorem riotassuni 7347
Description: The restricted iota class is limited in size by the base set. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotassuni (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotassuni
StepHypRef Expression
1 riotauni 7312 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 4036 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4873 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 4132 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 3952 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5eqsstrdi 3997 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 7346 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 4355 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8eqsstrdi 3997 . 2 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
106, 9pm2.61i 182 1 (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  ∃!wreu 3350  {crab 3406  cun 3907  wss 3909  c0 4281  𝒫 cpw 4559   cuni 4864  crio 7305
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 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-sn 4586  df-pr 4588  df-uni 4865  df-iota 6444  df-riota 7306
This theorem is referenced by: (None)
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