Theorem vsetrec 50501

Description: Construct V using set recursion. The proof indirectly uses trcl 9693, which relies on rec, but theoretically 𝐶 in trcl 9693 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 9712	. 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2		vex 3459	. . . 4 ⊢ 𝑎 ∈ V
3	2	pwid 4585	. . 3 ⊢ 𝑎 ∈ 𝒫 𝑎
4		pweq 4576	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5		vsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6		vpwex 5348	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
7	4, 5, 6	fvmpt 6989	. . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎)
8	2, 7	ax-mp 5	. . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎
9		eqid 2763	. . . . . 6 ⊢ setrecs(𝐹) = setrecs(𝐹)
10	2	a1i 11	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
11		id 23	. . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
12	9, 10, 11	setrec1 50489	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
13	8, 12	eqsstrrid 3976	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
14	13	sseld 3936	. . 3 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎 → 𝑎 ∈ setrecs(𝐹)))
15	3, 14	mpi 21	. 2 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
16	1, 15	mpg 1827	1 ⊢ setrecs(𝐹) = V

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562   ↦ cmpt 5192  ‘cfv 6536  setrecscsetrecs 50481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732  df-rank 9733  df-setrecs 50482

This theorem is referenced by: (None)