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Theorem 0setrec 47235
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 2733 . . 3 setrecs(𝐹) = setrecs(𝐹)
2 ss0 4359 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
3 fveq2 6843 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
4 0setrec.1 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
53, 4sylan9eqr 2795 . . . . . 6 ((𝜑𝑥 = ∅) → (𝐹𝑥) = ∅)
65ex 414 . . . . 5 (𝜑 → (𝑥 = ∅ → (𝐹𝑥) = ∅))
7 eqimss 4001 . . . . 5 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ ∅)
82, 6, 7syl56 36 . . . 4 (𝜑 → (𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
98alrimiv 1931 . . 3 (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
101, 9setrec2v 47227 . 2 (𝜑 → setrecs(𝐹) ⊆ ∅)
11 ss0 4359 . 2 (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
1210, 11syl 17 1 (𝜑 → setrecs(𝐹) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3911  c0 4283  cfv 6497  setrecscsetrecs 47214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-setrecs 47215
This theorem is referenced by: (None)
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