Theorem preddif 5705

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 3883	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋})) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 5680	. 2 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑋}))
3		df-pred 5680	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
4		df-pred 5680	. . 3 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
5	3, 4	difeq12i 3726	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋)) = ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∖ (𝐵 ∩ (◡𝑅 “ {𝑋})))
6	1, 2, 5	3eqtr4i 2654	1 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   = wceq 1483   ∖ cdif 3571   ∩ cin 3573  {csn 4177  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-pred 5680

This theorem is referenced by:  wfrlem8  7422