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Theorem bj-projex 33314
Description: Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projex (𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)

Proof of Theorem bj-projex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-bj-proj 33310 . 2 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
2 bj-clex 33283 . 2 (𝐵𝑉 → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} ∈ V)
31, 2syl5eqel 2854 1 (𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  {cab 2757  Vcvv 3351  {csn 4317  cima 5253   Proj bj-cproj 33309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-bj-proj 33310
This theorem is referenced by:  bj-pr1ex  33325  bj-pr2ex  33339
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