Theorem 0setrec 47702

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 2732	. . 3 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		ss0 4397	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
4		0setrec.1	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
5	3, 4	sylan9eqr 2794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅)
6	5	ex 413	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅))
7		eqimss 4039	. . . . 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 47694	. 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅)
11		ss0 4397	. 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
12	10, 11	syl 17	1 ⊢ (𝜑 → setrecs(𝐹) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3947  ∅c0 4321  ‘cfv 6540  setrecscsetrecs 47681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-setrecs 47682

This theorem is referenced by: (None)