Theorem vsetrec 50178

Description: Construct V using set recursion. The proof indirectly uses trcl 9649, which relies on rec, but theoretically 𝐶 in trcl 9649 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 9668	. 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2		vex 3433	. . . 4 ⊢ 𝑎 ∈ V
3	2	pwid 4563	. . 3 ⊢ 𝑎 ∈ 𝒫 𝑎
4		pweq 4555	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5		vsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6		vpwex 5319	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
7	4, 5, 6	fvmpt 6947	. . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎)
8	2, 7	ax-mp 5	. . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎
9		eqid 2736	. . . . . 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 50166	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
13	8, 12	eqsstrrid 3961	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
14	13	sseld 3920	. . 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 1799	1 ⊢ setrecs(𝐹) = V

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541   ↦ cmpt 5166  ‘cfv 6498  setrecscsetrecs 50158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-rank 9689  df-setrecs 50159

This theorem is referenced by: (None)