Theorem 0setrec 46409

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 2738	. . 3 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		ss0 4332	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
4		0setrec.1	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
5	3, 4	sylan9eqr 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅)
6	5	ex 413	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅))
7		eqimss 3977	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅)
8	2, 6, 7	syl56 36	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
9	8	alrimiv 1930	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
10	1, 9	setrec2v 46402	. 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅)
11		ss0 4332	. 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
12	10, 11	syl 17	1 ⊢ (𝜑 → setrecs(𝐹) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3887  ∅c0 4256  ‘cfv 6433  setrecscsetrecs 46389

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-setrecs 46390

This theorem is referenced by: (None)