Theorem preddif 6335

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 4239	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6307	. 2 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6307	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 6307	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	difeq12i 4060	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2793	1 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

This proof depends on syntax axioms:   = wceq 1569   ∖ cdif 4044   ∩ cin 4088  {csn 4584  ^◡ccnv 5650   “ cima 5654  Predcpred 6306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 4046  df-in 4090  df-pred 6307

This theorem is used by:  frrlem13  8324