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Theorem vsetrec 47234
Description: Construct V using set recursion. The proof indirectly uses trcl 9669, which relies on rec, but theoretically 𝐶 in trcl 9669 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 9675 . 2 (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2 vex 3448 . . . 4 𝑎 ∈ V
32pwid 4583 . . 3 𝑎 ∈ 𝒫 𝑎
4 pweq 4575 . . . . . . 7 (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5 vsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6 vpwex 5333 . . . . . . 7 𝒫 𝑎 ∈ V
74, 5, 6fvmpt 6949 . . . . . 6 (𝑎 ∈ V → (𝐹𝑎) = 𝒫 𝑎)
82, 7ax-mp 5 . . . . 5 (𝐹𝑎) = 𝒫 𝑎
9 eqid 2733 . . . . . 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 47222 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
138, 12eqsstrrid 3994 . . . 4 (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
1413sseld 3944 . . 3 (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎𝑎 ∈ setrecs(𝐹)))
153, 14mpi 20 . 2 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
161, 15mpg 1800 1 setrecs(𝐹) = V
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  𝒫 cpw 4561  cmpt 5189  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: (None)
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