Theorem 0setrec 48935

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.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		ss0 4408	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
4		0setrec.1	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
5	3, 4	sylan9eqr 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅)
6	5	ex 412	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅))
7		eqimss 4054	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅)
8	2, 6, 7	syl56 36	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
9	8	alrimiv 1925	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
10	1, 9	setrec2v 48927	. 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅)
11		ss0 4408	. 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
12	10, 11	syl 17	1 ⊢ (𝜑 → setrecs(𝐹) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3963  ∅c0 4339  ‘cfv 6563  setrecscsetrecs 48914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-setrecs 48915

This theorem is referenced by: (None)