Theorem preddif 6295

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 4249	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6267	. 2 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6267	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6267	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	difeq12i 4078	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2770	1 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1542   ∖ cdif 3900   ∩ cin 3902  {csn 4582  ^◡ccnv 5631   “ cima 5635  Predcpred 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-in 3910  df-pred 6267

This theorem is referenced by:  frrlem13  8250