Theorem 0setrec 50063

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 2737	. . 3 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		ss0 4356	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
4		0setrec.1	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
5	3, 4	sylan9eqr 2794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅)
6	5	ex 412	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅))
7		eqimss 3994	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅)
8	2, 6, 7	syl56 36	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
9	8	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
10	1, 9	setrec2v 50055	. 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅)
11		ss0 4356	. 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
12	10, 11	syl 17	1 ⊢ (𝜑 → setrecs(𝐹) = ∅)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3903  ∅c0 4287  ‘cfv 6500  setrecscsetrecs 50042

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-setrecs 50043

This theorem is referenced by: (None)