Theorem 0setrec 49690

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 2729	. . 3 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		ss0 4365	. . . . 5 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
3		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅))
4		0setrec.1	. . . . . . 7 ⊢ (𝜑 → (𝐹‘∅) = ∅)
5	3, 4	sylan9eqr 2786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝐹‘𝑥) = ∅)
6	5	ex 412	. . . . 5 ⊢ (𝜑 → (𝑥 = ∅ → (𝐹‘𝑥) = ∅))
7		eqimss 4005	. . . . 5 ⊢ ((𝐹‘𝑥) = ∅ → (𝐹‘𝑥) ⊆ ∅)
8	2, 6, 7	syl56 36	. . . 4 ⊢ (𝜑 → (𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
9	8	alrimiv 1927	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ ∅ → (𝐹‘𝑥) ⊆ ∅))
10	1, 9	setrec2v 49682	. 2 ⊢ (𝜑 → setrecs(𝐹) ⊆ ∅)
11		ss0 4365	. 2 ⊢ (setrecs(𝐹) ⊆ ∅ → setrecs(𝐹) = ∅)
12	10, 11	syl 17	1 ⊢ (𝜑 → setrecs(𝐹) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3914  ∅c0 4296  ‘cfv 6511  setrecscsetrecs 49669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-setrecs 49670

This theorem is referenced by: (None)