Theorem bj-projex 36956

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 36952	. 2 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2		bj-clexab 36925	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} ∈ V)
3	1, 2	eqeltrid 2832	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  {cab 2707  Vcvv 3444  {csn 4585   “ cima 5634   Proj bj-cproj 36951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-bj-proj 36952

This theorem is referenced by:  bj-pr1ex  36967  bj-pr2ex  36981