Theorem preddif 6330

Description: Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)

Assertion

Ref	Expression
preddif	⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem preddif

Step	Hyp	Ref	Expression
1		indifdir 4248	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6302	. 2 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6302	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6302	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	difeq12i 4079	. 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))

Syntax hints:   = wceq 1570   ∖ cdif 3902   ∩ cin 3904  {csn 4589  ^◡ccnv 5660   “ cima 5664  Predcpred 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-pred 6302

This theorem is referenced by:  frrlem13  8291