Theorem bj-projex 32176

Description: Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projex	⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)

Proof of Theorem bj-projex

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-proj 32172	. 2 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2		bj-clex 32145	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} ∈ V)
3	1, 2	syl5eqel 2692	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1977  {cab 2596  Vcvv 3173  {csn 4125   “ cima 5041   Proj bj-cproj 32171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-bj-proj 32172

This theorem is referenced by:  bj-pr1ex  32187  bj-pr2ex  32201