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Theorem 0setrec 48321
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 2725 . . 3 setrecs(𝐹) = setrecs(𝐹)
2 ss0 4400 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
3 fveq2 6896 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
4 0setrec.1 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
53, 4sylan9eqr 2787 . . . . . 6 ((𝜑𝑥 = ∅) → (𝐹𝑥) = ∅)
65ex 411 . . . . 5 (𝜑 → (𝑥 = ∅ → (𝐹𝑥) = ∅))
7 eqimss 4035 . . . . 5 ((𝐹𝑥) = ∅ → (𝐹𝑥) ⊆ ∅)
82, 6, 7syl56 36 . . . 4 (𝜑 → (𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
98alrimiv 1922 . . 3 (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹𝑥) ⊆ ∅))
101, 9setrec2v 48313 . 2 (𝜑 → setrecs(𝐹) ⊆ ∅)
11 ss0 4400 . 2 (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
1210, 11syl 17 1 (𝜑 → setrecs(𝐹) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wss 3944  c0 4322  cfv 6549  setrecscsetrecs 48300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-setrecs 48301
This theorem is referenced by: (None)
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