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Theorem riotassuni 7428
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 7394 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 4080 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4916 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 4179 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 3993 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5eqsstrdi 4028 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 7427 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 4400 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8eqsstrdi 4028 . 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 3378  {crab 3436  cun 3949  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-riota 7388
This theorem is referenced by: (None)
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