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Theorem vsetrec 48003
Description: Construct V using set recursion. The proof indirectly uses trcl 9722, which relies on rec, but theoretically 𝐶 in trcl 9722 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 9728 . 2 (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2 vex 3472 . . . 4 𝑎 ∈ V
32pwid 4619 . . 3 𝑎 ∈ 𝒫 𝑎
4 pweq 4611 . . . . . . 7 (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5 vsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6 vpwex 5368 . . . . . . 7 𝒫 𝑎 ∈ V
74, 5, 6fvmpt 6991 . . . . . 6 (𝑎 ∈ V → (𝐹𝑎) = 𝒫 𝑎)
82, 7ax-mp 5 . . . . 5 (𝐹𝑎) = 𝒫 𝑎
9 eqid 2726 . . . . . 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 47991 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
138, 12eqsstrrid 4026 . . . 4 (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
1413sseld 3976 . . 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 3468  wss 3943  𝒫 cpw 4597  cmpt 5224  cfv 6536  setrecscsetrecs 47983
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-r1 9758  df-rank 9759  df-setrecs 47984
This theorem is referenced by: (None)
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