Theorem bj-clexab 36930

Description: Sethood of certain classes. (Contributed by BJ, 2-Apr-2019.)

Assertion

Ref	Expression
bj-clexab	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-clexab

Step	Hyp	Ref	Expression
1		imaexg 7953	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
2		bj-snsetex 36929	. 2 ⊢ ((𝐴 “ 𝐵) ∈ V → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐵)} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  {cab 2717  Vcvv 3488  {csn 4648   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  bj-projex  36961