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Theorem riotassuni 7427
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 7393 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 4089 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4920 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 4188 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 4004 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5eqsstrdi 4049 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 7426 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 4405 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8eqsstrdi 4049 . 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 3375  {crab 3432  cun 3960  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911  crio 7386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-uni 4912  df-iota 6515  df-riota 7387
This theorem is referenced by: (None)
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