Theorem elsetrecs 48931

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 48922 and setrec2 48926, 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 48930	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
3		vex 3482	. . . . . 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 48922	. . . 4 ⊢ (𝑥 ⊆ 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
7	6	sselda 3995	. . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵)
8	7	exlimiv 1928	. 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)) → 𝐴 ∈ 𝐵)
9	2, 8	impbii 209	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  ‘cfv 6563  setrecscsetrecs 48914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803  df-setrecs 48915

This theorem is referenced by:  elpg  48945