Theorem bj-projex 37321

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 37317	. 2 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2		bj-clexab 37290	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} ∈ V)
3	1, 2	eqeltrid 2841	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  {cab 2715  Vcvv 3430  {csn 4568   “ cima 5628   Proj bj-cproj 37316

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 2709  ax-rep 5213  ax-sep 5232  ax-pr 5371  ax-un 7683

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 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-bj-proj 37317

This theorem is referenced by:  bj-pr1ex  37332  bj-pr2ex  37346