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Theorem 0setrec 42246
 Description: If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.)
Hypothesis
Ref Expression
0setrec.1 (𝜑 → (𝐹‘∅) = ∅)
Assertion
Ref Expression
0setrec (𝜑 → setrecs(𝐹) = ∅)

Proof of Theorem 0setrec
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 setrecs(𝐹) = setrecs(𝐹)
2 ss0 3926 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
3 fveq2 6103 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
4 0setrec.1 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
53, 4sylan9eqr 2666 . . . . . 6 ((𝜑𝑥 = ∅) → (𝐹𝑥) = ∅)
65ex 449 . . . . 5 (𝜑 → (𝑥 = ∅ → (𝐹𝑥) = ∅))
7 eqimss 3620 . . . . 5 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ ∅)
82, 6, 7syl56 35 . . . 4 (𝜑 → (𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
98alrimiv 1842 . . 3 (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
101, 9setrec2v 42242 . 2 (𝜑 → setrecs(𝐹) ⊆ ∅)
11 ss0 3926 . 2 (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
1210, 11syl 17 1 (𝜑 → setrecs(𝐹) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  setrecscsetrecs 42229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-setrecs 42230 This theorem is referenced by: (None)
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