Theorem elsetrecs 46383

Description: A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 46375 and setrec2 46379, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.)

Hypothesis

Ref	Expression
elsetrecs.1	⊢ 𝐵 = setrecs(𝐹)

Assertion

Ref	Expression
elsetrecs	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem elsetrecs

Step	Hyp	Ref	Expression
1		elsetrecs.1	. . 3 ⊢ 𝐵 = setrecs(𝐹)
2	1	elsetrecslem 46382	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
3		vex 3433	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	a1i 11	. . . . 5 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ∈ V)
5		id 22	. . . . 5 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐵)
6	1, 4, 5	setrec1 46375	. . . 4 ⊢ (𝑥 ⊆ 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
7	6	sselda 3920	. . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵)
8	7	exlimiv 1933	. 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵)
9	2, 8	impbii 208	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3429   ⊆ wss 3886  ‘cfv 6426  setrecscsetrecs 46367

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 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-reg 9338  ax-inf2 9386

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  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-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-om 7703  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-r1 9532  df-rank 9533  df-setrecs 46368

This theorem is referenced by:  elpg  46397