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Theorem vsetrec 42977
 Description: Construct V using set recursion. The proof indirectly uses trcl 8779, which relies on rec, but theoretically 𝐶 in trcl 8779 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 8785 . 2 (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2 vex 3343 . . . 4 𝑎 ∈ V
32pwid 4318 . . 3 𝑎 ∈ 𝒫 𝑎
4 pweq 4305 . . . . . . 7 (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5 vsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
62pwex 4997 . . . . . . 7 𝒫 𝑎 ∈ V
74, 5, 6fvmpt 6445 . . . . . 6 (𝑎 ∈ V → (𝐹𝑎) = 𝒫 𝑎)
82, 7ax-mp 5 . . . . 5 (𝐹𝑎) = 𝒫 𝑎
9 eqid 2760 . . . . . 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 42966 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
138, 12syl5eqssr 3791 . . . 4 (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
1413sseld 3743 . . 3 (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎𝑎 ∈ setrecs(𝐹)))
153, 14mpi 20 . 2 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
161, 15mpg 1873 1 setrecs(𝐹) = V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715  𝒫 cpw 4302   ↦ cmpt 4881  ‘cfv 6049  setrecscsetrecs 42958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-reg 8664  ax-inf2 8713 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-r1 8802  df-rank 8803  df-setrecs 42959 This theorem is referenced by: (None)
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