Theorem bj-clexab 37271

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 7864	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
2		bj-snsetex 37270	. 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 2114  {cab 2714  Vcvv 3429  {csn 4567   “ cima 5634

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-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  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-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  bj-projex  37302