Theorem vsetrec 49665

Description: Construct V using set recursion. The proof indirectly uses trcl 9657, which relies on rec, but theoretically 𝐶 in trcl 9657 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 9663	. 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2		vex 3448	. . . 4 ⊢ 𝑎 ∈ V
3	2	pwid 4581	. . 3 ⊢ 𝑎 ∈ 𝒫 𝑎
4		pweq 4573	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5		vsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6		vpwex 5327	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
7	4, 5, 6	fvmpt 6950	. . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎)
8	2, 7	ax-mp 5	. . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎
9		eqid 2729	. . . . . 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 49653	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
13	8, 12	eqsstrrid 3983	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
14	13	sseld 3942	. . 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 1797	1 ⊢ setrecs(𝐹) = V

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559   ↦ cmpt 5183  ‘cfv 6499  setrecscsetrecs 49645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-r1 9693  df-rank 9694  df-setrecs 49646

This theorem is referenced by: (None)