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Theorem 0setrec 43345
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 2825 . . 3 setrecs(𝐹) = setrecs(𝐹)
2 ss0 4199 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
3 fveq2 6433 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
4 0setrec.1 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
53, 4sylan9eqr 2883 . . . . . 6 ((𝜑𝑥 = ∅) → (𝐹𝑥) = ∅)
65ex 403 . . . . 5 (𝜑 → (𝑥 = ∅ → (𝐹𝑥) = ∅))
7 eqimss 3882 . . . . 5 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ ∅)
82, 6, 7syl56 36 . . . 4 (𝜑 → (𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
98alrimiv 2028 . . 3 (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
101, 9setrec2v 43338 . 2 (𝜑 → setrecs(𝐹) ⊆ ∅)
11 ss0 4199 . 2 (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
1210, 11syl 17 1 (𝜑 → setrecs(𝐹) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wss 3798  c0 4144  cfv 6123  setrecscsetrecs 43325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-setrecs 43326
This theorem is referenced by: (None)
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