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Theorem vsetrec 45623
 Description: Construct V using set recursion. The proof indirectly uses trcl 9203, which relies on rec, but theoretically 𝐶 in trcl 9203 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.)
Hypothesis
Ref Expression
vsetrec.1 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
Assertion
Ref Expression
vsetrec setrecs(𝐹) = V

Proof of Theorem vsetrec
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 setind 9209 . 2 (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2 vex 3413 . . . 4 𝑎 ∈ V
32pwid 4518 . . 3 𝑎 ∈ 𝒫 𝑎
4 pweq 4510 . . . . . . 7 (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5 vsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
62pwex 5249 . . . . . . 7 𝒫 𝑎 ∈ V
74, 5, 6fvmpt 6759 . . . . . 6 (𝑎 ∈ V → (𝐹𝑎) = 𝒫 𝑎)
82, 7ax-mp 5 . . . . 5 (𝐹𝑎) = 𝒫 𝑎
9 eqid 2758 . . . . . 6 setrecs(𝐹) = setrecs(𝐹)
102a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
11 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
129, 10, 11setrec1 45612 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
138, 12eqsstrrid 3941 . . . 4 (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
1413sseld 3891 . . 3 (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎𝑎 ∈ setrecs(𝐹)))
153, 14mpi 20 . 2 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
161, 15mpg 1799 1 setrecs(𝐹) = V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494   ↦ cmpt 5112  ‘cfv 6335  setrecscsetrecs 45604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-reg 9089  ax-inf2 9137 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-r1 9226  df-rank 9227  df-setrecs 45605 This theorem is referenced by: (None)
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