Theorem preddif 6187

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 4199	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6160	. 2 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6160	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6160	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	difeq12i 4035	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2775	1 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1543   ∖ cdif 3863   ∩ cin 3865  {csn 4541  ^◡ccnv 5550   “ cima 5554  Predcpred 6159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-in 3873  df-pred 6160

This theorem is referenced by:  frrlem13  8039  wfrlem8  8062