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Theorem riotassuni 6603
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 6572 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 3671 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4432 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 3760 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 3597 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5syl6eqss 3639 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 6602 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 3949 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8syl6eqss 3639 . 2 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
106, 9pm2.61i 176 1 (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  ∃!wreu 2914  {crab 2916  cun 3558  wss 3560  c0 3896  𝒫 cpw 4135   cuni 4407  crio 6565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5813  df-riota 6566
This theorem is referenced by: (None)
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