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Theorem elsetrecs 47231
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 47222 and setrec2 47226, 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 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem elsetrecs
StepHypRef Expression
1 elsetrecs.1 . . 3 𝐵 = setrecs(𝐹)
21elsetrecslem 47230 . 2 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
3 vex 3448 . . . . . 6 𝑥 ∈ V
43a1i 11 . . . . 5 (𝑥𝐵𝑥 ∈ V)
5 id 22 . . . . 5 (𝑥𝐵𝑥𝐵)
61, 4, 5setrec1 47222 . . . 4 (𝑥𝐵 → (𝐹𝑥) ⊆ 𝐵)
76sselda 3945 . . 3 ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐴𝐵)
87exlimiv 1934 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐴𝐵)
92, 8impbii 208 1 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  wss 3911  cfv 6497  setrecscsetrecs 47214
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-r1 9705  df-rank 9706  df-setrecs 47215
This theorem is referenced by:  elpg  47245
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