Theorem predin 6285

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 4171	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6259	. 2 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6259	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6259	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	ineq12i 4154	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2773	1 ⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1547   ∩ cin 3889  {csn 4562  ^◡ccnv 5624   “ cima 5628  Predcpred 6258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-in 3897  df-pred 6259

This theorem is referenced by: (None)