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Metamath Proof Explorer

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 ) )