Theorem predasetex 5612

Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)

Hypothesis

Ref	Expression
predasetex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
predasetex	⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V

Proof of Theorem predasetex

Step	Hyp	Ref	Expression
1		df-pred 5597	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		predasetex.1	. . 3 ⊢ 𝐴 ∈ V
3	2	inex1 4727	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V
4	1, 3	eqeltri 2684	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V

Syntax hints:   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  {csn 4125  ^◡ccnv 5037   “ cima 5041  Predcpred 5596

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-pred 5597

This theorem is referenced by: (None)