Theorem predexg 6273

Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)

Assertion

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

Proof of Theorem predexg

Step	Hyp	Ref	Expression
1		df-pred 6255	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inex1g 5261	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V)
3	1, 2	eqeltrid 2837	1 ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  {csn 4577  ^◡ccnv 5620   “ cima 5624  Predcpred 6254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-in 3905  df-pred 6255

This theorem is referenced by: (None)