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Theorem vsetrec 48212
Description: Construct V using set recursion. The proof indirectly uses trcl 9759, which relies on rec, but theoretically 𝐶 in trcl 9759 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 9765 . 2 (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2 vex 3477 . . . 4 𝑎 ∈ V
32pwid 4628 . . 3 𝑎 ∈ 𝒫 𝑎
4 pweq 4620 . . . . . . 7 (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5 vsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6 vpwex 5381 . . . . . . 7 𝒫 𝑎 ∈ V
74, 5, 6fvmpt 7010 . . . . . 6 (𝑎 ∈ V → (𝐹𝑎) = 𝒫 𝑎)
82, 7ax-mp 5 . . . . 5 (𝐹𝑎) = 𝒫 𝑎
9 eqid 2728 . . . . . 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 48200 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
138, 12eqsstrrid 4031 . . . 4 (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
1413sseld 3981 . . 3 (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎𝑎 ∈ setrecs(𝐹)))
153, 14mpi 20 . 2 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
161, 15mpg 1791 1 setrecs(𝐹) = V
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  wss 3949  𝒫 cpw 4606  cmpt 5235  cfv 6553  setrecscsetrecs 48192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-r1 9795  df-rank 9796  df-setrecs 48193
This theorem is referenced by: (None)
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