Theorem vsetrec 49814

Description: Construct V using set recursion. The proof indirectly uses trcl 9618, which relies on rec, but theoretically 𝐶 in trcl 9618 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.

Step	Hyp	Ref	Expression
1		setind 9637	. 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2		vex 3440	. . . 4 ⊢ 𝑎 ∈ V
3	2	pwid 4569	. . 3 ⊢ 𝑎 ∈ 𝒫 𝑎
4		pweq 4561	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5		vsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6		vpwex 5313	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
7	4, 5, 6	fvmpt 6929	. . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎)
8	2, 7	ax-mp 5	. . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎
9		eqid 2731	. . . . . 6 ⊢ setrecs(𝐹) = setrecs(𝐹)
10	2	a1i 11	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
11		id 22	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
12	9, 10, 11	setrec1 49802	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
13	8, 12	eqsstrrid 3969	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
14	13	sseld 3928	. . 3 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎 → 𝑎 ∈ setrecs(𝐹)))
15	3, 14	mpi 20	. 2 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
16	1, 15	mpg 1798	1 ⊢ setrecs(𝐹) = V

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547   ↦ cmpt 5170  ‘cfv 6481  setrecscsetrecs 49794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-reg 9478  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-r1 9657  df-rank 9658  df-setrecs 49795

This theorem is referenced by: (None)