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Theorem riotassuni 6790
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 6759 . . 3 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
2 ssrab2 3834 . . . . 5 {𝑥𝐴𝜑} ⊆ 𝐴
32unissi 4595 . . . 4 {𝑥𝐴𝜑} ⊆ 𝐴
4 ssun2 3926 . . . 4 𝐴 ⊆ (𝒫 𝐴 𝐴)
53, 4sstri 3759 . . 3 {𝑥𝐴𝜑} ⊆ (𝒫 𝐴 𝐴)
61, 5syl6eqss 3802 . 2 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) ⊆ (𝒫 𝐴 𝐴))
7 riotaund 6789 . . 3 (¬ ∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = ∅)
8 0ss 4114 . . 3 ∅ ⊆ (𝒫 𝐴 𝐴)
97, 8syl6eqss 3802 . 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 3062  {crab 3064  cun 3719  wss 3721  c0 4061  𝒫 cpw 4295   cuni 4572  crio 6752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-sn 4315  df-pr 4317  df-uni 4573  df-iota 5994  df-riota 6753
This theorem is referenced by: (None)
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