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Theorem riotassuni 7146
 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 7112 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 4054 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4853 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 4147 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 3974 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5eqsstrdi 4019 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 7145 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 4348 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8eqsstrdi 4019 . 2 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
106, 9pm2.61i 184 1 (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∃!wreu 3138  {crab 3140   ∪ cun 3932   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  ∪ cuni 4830  ℩crio 7105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562  df-uni 4831  df-iota 6307  df-riota 7106 This theorem is referenced by: (None)
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