Theorem vsetrec 50324

Description: Construct V using set recursion. The proof indirectly uses trcl 9683, which relies on rec, but theoretically 𝐶 in trcl 9683 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 9702	. 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V)
2		vex 3458	. . . 4 ⊢ 𝑎 ∈ V
3	2	pwid 4578	. . 3 ⊢ 𝑎 ∈ 𝒫 𝑎
4		pweq 4569	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎)
5		vsetrec.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)
6		vpwex 5334	. . . . . . 7 ⊢ 𝒫 𝑎 ∈ V
7	4, 5, 6	fvmpt 6975	. . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎)
8	2, 7	ax-mp 5	. . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎
9		eqid 2762	. . . . . 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 50312	. . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹))
13	8, 12	eqsstrrid 3975	. . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹))
14	13	sseld 3935	. . 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 1817	1 ⊢ setrecs(𝐹) = V

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555   ↦ cmpt 5181  ‘cfv 6521  setrecscsetrecs 50304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-reg 9540  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-r1 9722  df-rank 9723  df-setrecs 50305

This theorem is referenced by: (None)