Theorem predin 6356

Description: Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Assertion

Ref	Expression
predin	⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predin

Step	Hyp	Ref	Expression
1		inindir 4135	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6330	. 2 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6330	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6330	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	ineq12i 4118	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2796	1 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 4092  {csn 4588  ^◡ccnv 5657   “ cima 5661  Predcpred 6329

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-in 4094  df-pred 6330

This theorem is used by: (None)