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Theorem elsetrecs 44730
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 44722 and setrec2 44726, 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 44729 . 2 (𝐴𝐵 → ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
3 vex 3495 . . . . . 6 𝑥 ∈ V
43a1i 11 . . . . 5 (𝑥𝐵𝑥 ∈ V)
5 id 22 . . . . 5 (𝑥𝐵𝑥𝐵)
61, 4, 5setrec1 44722 . . . 4 (𝑥𝐵 → (𝐹𝑥) ⊆ 𝐵)
76sselda 3964 . . 3 ((𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐴𝐵)
87exlimiv 1922 . 2 (∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)) → 𝐴𝐵)
92, 8impbii 210 1 (𝐴𝐵 ↔ ∃𝑥(𝑥𝐵𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  wss 3933  cfv 6348  setrecscsetrecs 44714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181  df-rank 9182  df-setrecs 44715
This theorem is referenced by:  elpg  44744
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