Theorem bj-projex 35185

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 35181	. 2 ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2		bj-clex 35154	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} ∈ V)
3	1, 2	eqeltrid 2843	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 Proj 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  {cab 2715  Vcvv 3432  {csn 4561   “ cima 5592   Proj bj-cproj 35180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-bj-proj 35181

This theorem is referenced by:  bj-pr1ex  35196  bj-pr2ex  35210