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Theorem 0setrec 42960
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 2760 . . 3 setrecs(𝐹) = setrecs(𝐹)
2 ss0 4117 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
3 fveq2 6352 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
4 0setrec.1 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
53, 4sylan9eqr 2816 . . . . . 6 ((𝜑𝑥 = ∅) → (𝐹𝑥) = ∅)
65ex 449 . . . . 5 (𝜑 → (𝑥 = ∅ → (𝐹𝑥) = ∅))
7 eqimss 3798 . . . . 5 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ ∅)
82, 6, 7syl56 36 . . . 4 (𝜑 → (𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
98alrimiv 2004 . . 3 (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
101, 9setrec2v 42953 . 2 (𝜑 → setrecs(𝐹) ⊆ ∅)
11 ss0 4117 . 2 (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
1210, 11syl 17 1 (𝜑 → setrecs(𝐹) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wss 3715  c0 4058  cfv 6049  setrecscsetrecs 42940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-setrecs 42941
This theorem is referenced by: (None)
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