Theorem predin 6219

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 4158	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6191	. 2 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6191	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6191	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	ineq12i 4141	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2776	1 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1539   ∩ cin 3882  {csn 4558  ^◡ccnv 5579   “ cima 5583  Predcpred 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-pred 6191

This theorem is referenced by: (None)