Theorem preddif

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

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

Proof

Step	Hyp	Ref	Expression
1		indifdir	⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) = ( ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ∖ ( 𝐵 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) )
2		df-pred	⊢ Pred ( 𝑅 , ( 𝐴 ∖ 𝐵 ) , 𝑋 ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
3		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
4		df-pred	⊢ Pred ( 𝑅 , 𝐵 , 𝑋 ) = ( 𝐵 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
5	3 4	difeq12i	⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∖ Pred ( 𝑅 , 𝐵 , 𝑋 ) ) = ( ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ∖ ( 𝐵 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) )
6	1 2 5	3eqtr4i	⊢ Pred ( 𝑅 , ( 𝐴 ∖ 𝐵 ) , 𝑋 ) = ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∖ Pred ( 𝑅 , 𝐵 , 𝑋 ) )